10 5

10_4

10_6

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 10 5's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 5 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X3,12,4,13 X13,1,14,20 X5,15,6,14 X7,17,8,16 X9,19,10,18 X15,7,16,6 X17,9,18,8 X19,11,20,10 X11,2,12,3
Gauss code	-1, 10, -2, 1, -4, 7, -5, 8, -6, 9, -10, 2, -3, 4, -7, 5, -8, 6, -9, 3
Dowker-Thistlethwaite code	4 12 14 16 18 2 20 6 8 10
Conway Notation	[6112]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 10, width is 3, Braid index is 3		[{12, 8}, {1, 10}, {9, 11}, {10, 12}, {11, 7}, {8, 6}, {7, 5}, {6, 4}, {5, 3}, {4, 2}, {3, 1}, {2, 9}]

[edit Notes on presentations of 10 5]

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.org/wiki/KnotTheory.

In[3]:=

K = Knot["10 5"];

In[4]:=

PD[K]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

X1425 X3,12,4,13 X13,1,14,20 X5,15,6,14 X7,17,8,16 X9,19,10,18 X15,7,16,6 X17,9,18,8 X19,11,20,10 X11,2,12,3

In[5]:=

GaussCode[K]

Out[5]=

-1, 10, -2, 1, -4, 7, -5, 8, -6, 9, -10, 2, -3, 4, -7, 5, -8, 6, -9, 3

In[6]:=

DTCode[K]

Out[6]=

4 12 14 16 18 2 20 6 8 10

(The path below may be different on your system)

In[7]:=

AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:=

ConwayNotation[K]

Out[8]=

[6112]

In[9]:=

br = BR[K]

KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.

Out[9]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textrm{BR}(3,\{1,1,1,1,1,1,-2,1,-2,-2\})}

In[10]:=

{First[br], Crossings[br], BraidIndex[K]}

KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.

KnotTheory::loading: Loading precomputed data in IndianaData`.

Out[10]=

{ 3, 10, 3 }

In[11]:=

Show[BraidPlot[br]]

Out[11]=

-Graphics-

In[12]:=

Show[DrawMorseLink[K]]

KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."

KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

Out[12]=

-Graphics-

In[13]:=

ap = ArcPresentation[K]

Out[13]=

ArcPresentation[{12, 8}, {1, 10}, {9, 11}, {10, 12}, {11, 7}, {8, 6}, {7, 5}, {6, 4}, {5, 3}, {4, 2}, {3, 1}, {2, 9}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 4 Bridge index 2 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [0][-12] Hyperbolic Volume 7.37394 A-Polynomial See Data:10 5/A-polynomial

[edit Notes for 10 5's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2} Topological 4 genus $2$ Concordance genus $4$ Rasmussen s-Invariant 4

[edit Notes for 10 5's four dimensional invariants]

Polynomial invariants

Alexander polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^4-3 t^3+5 t^2-5 t+5-5 t^{-1} +5 t^{-2} -3 t^{-3} + t^{-4} }
Conway polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^8+5 z^6+7 z^4+4 z^2+1}
2nd Alexander ideal (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}
Determinant and Signature	{ 33, 4 }
Jones polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^9+2 q^8-3 q^7+4 q^6-5 q^5+5 q^4-4 q^3+4 q^2-2 q+2- q^{-1} }
HOMFLY-PT polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^8 a^{-4} -z^6 a^{-2} +7 z^6 a^{-4} -z^6 a^{-6} -5 z^4 a^{-2} +17 z^4 a^{-4} -5 z^4 a^{-6} -6 z^2 a^{-2} +17 z^2 a^{-4} -7 z^2 a^{-6} - a^{-2} +5 a^{-4} -3 a^{-6} }
Kauffman polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^9 a^{-3} +z^9 a^{-5} +2 z^8 a^{-2} +4 z^8 a^{-4} +2 z^8 a^{-6} +z^7 a^{-1} -3 z^7 a^{-3} -2 z^7 a^{-5} +2 z^7 a^{-7} -11 z^6 a^{-2} -20 z^6 a^{-4} -7 z^6 a^{-6} +2 z^6 a^{-8} -5 z^5 a^{-1} -2 z^5 a^{-3} -3 z^5 a^{-5} -4 z^5 a^{-7} +2 z^5 a^{-9} +18 z^4 a^{-2} +32 z^4 a^{-4} +10 z^4 a^{-6} -2 z^4 a^{-8} +2 z^4 a^{-10} +6 z^3 a^{-1} +7 z^3 a^{-3} +6 z^3 a^{-5} +3 z^3 a^{-7} -z^3 a^{-9} +z^3 a^{-11} -10 z^2 a^{-2} -22 z^2 a^{-4} -9 z^2 a^{-6} +z^2 a^{-8} -2 z^2 a^{-10} -z a^{-1} -2 z a^{-3} -3 z a^{-5} -z a^{-7} -z a^{-11} + a^{-2} +5 a^{-4} +3 a^{-6} }
The A2 invariant	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^2+ q^{-4} +2 q^{-6} + q^{-8} +2 q^{-10} - q^{-12} + q^{-14} - q^{-22} - q^{-26} }
The G2 invariant	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{12}-q^{10}+2 q^8-3 q^6+q^4-2 q^2-1+5 q^{-2} -8 q^{-4} +8 q^{-6} -7 q^{-8} +2 q^{-10} +3 q^{-12} -9 q^{-14} +10 q^{-16} -9 q^{-18} +5 q^{-20} +2 q^{-22} -4 q^{-24} +7 q^{-26} -3 q^{-28} +2 q^{-30} +4 q^{-32} -2 q^{-34} +2 q^{-36} +2 q^{-38} -2 q^{-40} +8 q^{-42} -4 q^{-44} +6 q^{-46} -2 q^{-48} - q^{-50} +7 q^{-52} -10 q^{-54} +9 q^{-56} -6 q^{-58} +2 q^{-60} +2 q^{-62} -6 q^{-64} +6 q^{-66} -5 q^{-68} +2 q^{-70} -3 q^{-74} + q^{-76} + q^{-78} -2 q^{-80} + q^{-82} - q^{-86} +2 q^{-88} -2 q^{-90} + q^{-92} + q^{-96} - q^{-98} - q^{-100} - q^{-104} +2 q^{-106} -4 q^{-108} +4 q^{-110} -3 q^{-112} - q^{-114} +2 q^{-116} -5 q^{-118} +5 q^{-120} -5 q^{-122} +3 q^{-124} - q^{-126} -2 q^{-128} +4 q^{-130} -4 q^{-132} +4 q^{-134} -2 q^{-136} + q^{-138} -2 q^{-142} +2 q^{-144} - q^{-146} + q^{-148} }

Further Quantum Invariants

Further quantum knot invariants for 10_5.

A1 Invariants.

Weight	Invariant
1	$-q^{3}+q+2q^{-3}+q^{-7}-q^{-11}+q^{-13}-q^{-15}+q^{-17}-q^{-19}$
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{12}-q^{10}-2 q^8+2 q^6-3 q^2+2+3 q^{-2} -2 q^{-4} + q^{-6} +3 q^{-8} +2 q^{-14} -2 q^{-18} + q^{-22} - q^{-24} - q^{-26} + q^{-28} -2 q^{-32} + q^{-34} - q^{-38} +2 q^{-40} - q^{-42} - q^{-44} +2 q^{-46} - q^{-48} - q^{-50} + q^{-52} }
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{27}+q^{25}+2 q^{23}-3 q^{19}-2 q^{17}+4 q^{15}+3 q^{13}-4 q^{11}-6 q^9+q^7+6 q^5+2 q^3-6 q-2 q^{-1} +4 q^{-3} +5 q^{-5} - q^{-7} -2 q^{-9} +3 q^{-13} +3 q^{-15} +2 q^{-17} -2 q^{-19} -2 q^{-21} +2 q^{-23} +4 q^{-25} -2 q^{-27} -5 q^{-29} +2 q^{-31} +3 q^{-33} -3 q^{-35} -6 q^{-37} +3 q^{-39} +4 q^{-41} -5 q^{-45} - q^{-47} +2 q^{-49} +3 q^{-51} + q^{-53} -5 q^{-55} -4 q^{-57} +4 q^{-59} +6 q^{-61} - q^{-63} -7 q^{-65} - q^{-67} +7 q^{-69} +3 q^{-71} -5 q^{-73} -4 q^{-75} +3 q^{-77} +5 q^{-79} -2 q^{-81} -4 q^{-83} +3 q^{-87} -2 q^{-91} + q^{-95} + q^{-97} - q^{-99} }
4	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{48}-q^{46}-2 q^{44}+q^{40}+5 q^{38}-4 q^{34}-4 q^{32}-3 q^{30}+10 q^{28}+7 q^{26}-2 q^{24}-8 q^{22}-13 q^{20}+4 q^{18}+10 q^{16}+8 q^{14}-16 q^{10}-6 q^8+3 q^6+10 q^4+9 q^2-6-5 q^{-2} -4 q^{-4} + q^{-6} +6 q^{-8} +5 q^{-12} +3 q^{-14} -2 q^{-16} -4 q^{-18} -7 q^{-20} +8 q^{-22} +15 q^{-24} +4 q^{-26} -9 q^{-28} -18 q^{-30} + q^{-32} +20 q^{-34} +11 q^{-36} -8 q^{-38} -22 q^{-40} -8 q^{-42} +16 q^{-44} +13 q^{-46} -3 q^{-48} -17 q^{-50} -8 q^{-52} +13 q^{-54} +10 q^{-56} -3 q^{-58} -12 q^{-60} -5 q^{-62} +11 q^{-64} +9 q^{-66} -2 q^{-68} -12 q^{-70} -9 q^{-72} +5 q^{-74} +12 q^{-76} +8 q^{-78} - q^{-80} -14 q^{-82} -15 q^{-84} +4 q^{-86} +19 q^{-88} +20 q^{-90} -8 q^{-92} -28 q^{-94} -13 q^{-96} +14 q^{-98} +33 q^{-100} +5 q^{-102} -27 q^{-104} -22 q^{-106} +3 q^{-108} +30 q^{-110} +10 q^{-112} -17 q^{-114} -18 q^{-116} -3 q^{-118} +22 q^{-120} +9 q^{-122} -8 q^{-124} -11 q^{-126} -6 q^{-128} +12 q^{-130} +5 q^{-132} -2 q^{-134} -5 q^{-136} -5 q^{-138} +6 q^{-140} + q^{-142} - q^{-146} -2 q^{-148} +3 q^{-150} - q^{-156} - q^{-158} + q^{-160} }
5	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{75}+q^{73}+2 q^{71}-q^{67}-3 q^{65}-3 q^{63}+6 q^{59}+6 q^{57}+q^{55}-5 q^{53}-10 q^{51}-8 q^{49}+3 q^{47}+16 q^{45}+15 q^{43}+3 q^{41}-11 q^{39}-22 q^{37}-15 q^{35}+5 q^{33}+22 q^{31}+21 q^{29}+7 q^{27}-13 q^{25}-28 q^{23}-18 q^{21}+3 q^{19}+20 q^{17}+22 q^{15}+8 q^{13}-11 q^{11}-18 q^9-12 q^7+q^5+11 q^3+8 q- q^{-1} - q^{-3} +2 q^{-5} +6 q^{-7} +3 q^{-9} -8 q^{-11} -20 q^{-13} -11 q^{-15} +15 q^{-17} +35 q^{-19} +27 q^{-21} -3 q^{-23} -39 q^{-25} -45 q^{-27} -7 q^{-29} +42 q^{-31} +59 q^{-33} +27 q^{-35} -31 q^{-37} -64 q^{-39} -44 q^{-41} +15 q^{-43} +62 q^{-45} +54 q^{-47} - q^{-49} -53 q^{-51} -60 q^{-53} -19 q^{-55} +40 q^{-57} +59 q^{-59} +27 q^{-61} -28 q^{-63} -55 q^{-65} -35 q^{-67} +14 q^{-69} +47 q^{-71} +37 q^{-73} -6 q^{-75} -38 q^{-77} -32 q^{-79} +2 q^{-81} +29 q^{-83} +30 q^{-85} -4 q^{-87} -29 q^{-89} -20 q^{-91} +5 q^{-93} +25 q^{-95} +21 q^{-97} -8 q^{-99} -29 q^{-101} -20 q^{-103} +6 q^{-105} +27 q^{-107} +27 q^{-109} +8 q^{-111} -19 q^{-113} -34 q^{-115} -25 q^{-117} +3 q^{-119} +35 q^{-121} +44 q^{-123} +22 q^{-125} -25 q^{-127} -59 q^{-129} -44 q^{-131} +10 q^{-133} +59 q^{-135} +59 q^{-137} +9 q^{-139} -53 q^{-141} -67 q^{-143} -21 q^{-145} +41 q^{-147} +61 q^{-149} +24 q^{-151} -30 q^{-153} -50 q^{-155} -21 q^{-157} +26 q^{-159} +40 q^{-161} +13 q^{-163} -25 q^{-165} -30 q^{-167} -4 q^{-169} +22 q^{-171} +25 q^{-173} + q^{-175} -23 q^{-177} -22 q^{-179} +2 q^{-181} +19 q^{-183} +17 q^{-185} +2 q^{-187} -13 q^{-189} -15 q^{-191} -3 q^{-193} +8 q^{-195} +9 q^{-197} +3 q^{-199} -3 q^{-201} -5 q^{-203} -3 q^{-205} + q^{-207} +3 q^{-209} + q^{-211} - q^{-217} + q^{-219} - q^{-223} - q^{-225} + q^{-231} + q^{-233} - q^{-235} }
6	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{108}-q^{106}-2 q^{104}+q^{100}+3 q^{98}+q^{96}+3 q^{94}-2 q^{92}-8 q^{90}-5 q^{88}-q^{86}+6 q^{84}+7 q^{82}+14 q^{80}+3 q^{78}-12 q^{76}-18 q^{74}-17 q^{72}-3 q^{70}+6 q^{68}+33 q^{66}+30 q^{64}+8 q^{62}-15 q^{60}-35 q^{58}-34 q^{56}-27 q^{54}+19 q^{52}+43 q^{50}+45 q^{48}+25 q^{46}-8 q^{44}-36 q^{42}-62 q^{40}-30 q^{38}+3 q^{36}+37 q^{34}+49 q^{32}+36 q^{30}+8 q^{28}-38 q^{26}-40 q^{24}-33 q^{22}-3 q^{20}+22 q^{18}+31 q^{16}+22 q^{14}-10 q^{12}-14 q^{10}-17 q^8+6 q^6+20 q^4+17 q^2-2-40 q^{-2} -43 q^{-4} -26 q^{-6} +31 q^{-8} +74 q^{-10} +75 q^{-12} +32 q^{-14} -53 q^{-16} -107 q^{-18} -109 q^{-20} -25 q^{-22} +77 q^{-24} +142 q^{-26} +136 q^{-28} +29 q^{-30} -94 q^{-32} -173 q^{-34} -136 q^{-36} -23 q^{-38} +105 q^{-40} +189 q^{-42} +143 q^{-44} +18 q^{-46} -122 q^{-48} -178 q^{-50} -139 q^{-52} -22 q^{-54} +122 q^{-56} +177 q^{-58} +131 q^{-60} +5 q^{-62} -108 q^{-64} -170 q^{-66} -137 q^{-68} -4 q^{-70} +114 q^{-72} +165 q^{-74} +113 q^{-76} +6 q^{-78} -119 q^{-80} -176 q^{-82} -100 q^{-84} +26 q^{-86} +133 q^{-88} +152 q^{-90} +85 q^{-92} -52 q^{-94} -157 q^{-96} -132 q^{-98} -31 q^{-100} +85 q^{-102} +138 q^{-104} +105 q^{-106} -10 q^{-108} -116 q^{-110} -113 q^{-112} -41 q^{-114} +49 q^{-116} +94 q^{-118} +75 q^{-120} -4 q^{-122} -76 q^{-124} -67 q^{-126} -12 q^{-128} +46 q^{-130} +61 q^{-132} +30 q^{-134} -30 q^{-136} -67 q^{-138} -45 q^{-140} +9 q^{-142} +60 q^{-144} +70 q^{-146} +38 q^{-148} -22 q^{-150} -68 q^{-152} -83 q^{-154} -48 q^{-156} +16 q^{-158} +77 q^{-160} +103 q^{-162} +75 q^{-164} +7 q^{-166} -95 q^{-168} -143 q^{-170} -115 q^{-172} -11 q^{-174} +106 q^{-176} +177 q^{-178} +153 q^{-180} +8 q^{-182} -130 q^{-184} -206 q^{-186} -148 q^{-188} -10 q^{-190} +145 q^{-192} +220 q^{-194} +132 q^{-196} -9 q^{-198} -152 q^{-200} -182 q^{-202} -116 q^{-204} +25 q^{-206} +142 q^{-208} +137 q^{-210} +79 q^{-212} -31 q^{-214} -89 q^{-216} -107 q^{-218} -53 q^{-220} +22 q^{-222} +46 q^{-224} +70 q^{-226} +41 q^{-228} +17 q^{-230} -32 q^{-232} -49 q^{-234} -40 q^{-236} -34 q^{-238} +17 q^{-240} +40 q^{-242} +57 q^{-244} +22 q^{-246} -11 q^{-248} -35 q^{-250} -53 q^{-252} -18 q^{-254} +8 q^{-256} +41 q^{-258} +30 q^{-260} +14 q^{-262} -6 q^{-264} -31 q^{-266} -18 q^{-268} -12 q^{-270} +11 q^{-272} +13 q^{-274} +13 q^{-276} +7 q^{-278} -7 q^{-280} -4 q^{-282} -11 q^{-284} - q^{-286} +5 q^{-290} +5 q^{-292} +3 q^{-296} -5 q^{-298} - q^{-300} -2 q^{-302} +3 q^{-310} - q^{-312} + q^{-314} - q^{-320} - q^{-322} + q^{-324} }

A2 Invariants.

Weight	Invariant
1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^2+ q^{-4} +2 q^{-6} + q^{-8} +2 q^{-10} - q^{-12} + q^{-14} - q^{-22} - q^{-26} }
1,1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{12}-2 q^{10}+4 q^8-8 q^6+13 q^4-20 q^2+24-32 q^{-2} +30 q^{-4} -30 q^{-6} +24 q^{-8} -10 q^{-10} +6 q^{-12} +18 q^{-14} -18 q^{-16} +36 q^{-18} -36 q^{-20} +44 q^{-22} -44 q^{-24} +38 q^{-26} -35 q^{-28} +22 q^{-30} -20 q^{-32} +6 q^{-34} -4 q^{-36} -2 q^{-38} +2 q^{-40} + q^{-44} +2 q^{-48} -4 q^{-50} +5 q^{-52} -10 q^{-54} +12 q^{-56} -12 q^{-58} +13 q^{-60} -12 q^{-62} +12 q^{-64} -10 q^{-66} +7 q^{-68} -6 q^{-70} +4 q^{-72} -2 q^{-74} + q^{-76} }
2,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{10}-q^6-q^4-1-3 q^{-2} +2 q^{-6} + q^{-8} +4 q^{-12} +4 q^{-14} +3 q^{-16} +2 q^{-18} +3 q^{-20} - q^{-24} - q^{-30} + q^{-34} -3 q^{-36} -3 q^{-38} - q^{-40} - q^{-44} - q^{-46} + q^{-48} + q^{-50} - q^{-54} + q^{-66} }

A3 Invariants.

Weight	Invariant
0,1,0	$q^{6}-q^{4}-3q^{-2}+2q^{-10}+3q^{-12}+q^{-14}+4q^{-16}+2q^{-18}+q^{-20}+4q^{-22}+2q^{-24}-q^{-26}-q^{-30}-3q^{-32}-3q^{-34}-q^{-36}-2q^{-40}+2q^{-44}-2q^{-46}-q^{-48}+3q^{-50}-q^{-52}-2q^{-54}+2q^{-56}-q^{-60}+q^{-62}$
1,0,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q- q^{-3} + q^{-5} +3 q^{-9} + q^{-11} +2 q^{-13} + q^{-15} + q^{-17} + q^{-19} + q^{-23} -2 q^{-25} -2 q^{-29} - q^{-33} }
1,0,1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{12}-2 q^{10}+3 q^8-3 q^6+q^4+3 q^2-6+9 q^{-2} -12 q^{-4} +3 q^{-6} -6 q^{-8} -7 q^{-10} +3 q^{-12} -6 q^{-14} +10 q^{-16} - q^{-18} +21 q^{-20} - q^{-22} +26 q^{-24} +2 q^{-26} +11 q^{-28} +14 q^{-30} -16 q^{-32} +17 q^{-34} -23 q^{-36} +7 q^{-38} -20 q^{-40} -10 q^{-44} -10 q^{-46} +7 q^{-48} -19 q^{-50} +17 q^{-52} -12 q^{-54} +9 q^{-56} +3 q^{-58} -2 q^{-60} +6 q^{-62} -2 q^{-64} +2 q^{-66} -5 q^{-68} +6 q^{-70} -8 q^{-72} +4 q^{-74} +2 q^{-76} -7 q^{-78} +11 q^{-80} -7 q^{-82} +2 q^{-84} +4 q^{-86} -8 q^{-88} +6 q^{-90} -3 q^{-92} - q^{-94} +3 q^{-96} -2 q^{-98} + q^{-100} }

A4 Invariants.

Weight	Invariant
0,1,0,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^4- q^{-4} -3 q^{-6} -2 q^{-8} - q^{-10} - q^{-12} + q^{-16} +4 q^{-18} +3 q^{-20} +4 q^{-22} +4 q^{-24} +7 q^{-26} +6 q^{-28} +6 q^{-30} +3 q^{-32} +4 q^{-34} + q^{-36} -4 q^{-38} -4 q^{-40} -4 q^{-42} -6 q^{-44} -5 q^{-46} - q^{-48} -3 q^{-50} -2 q^{-52} + q^{-54} + q^{-56} -2 q^{-58} - q^{-60} +2 q^{-62} -2 q^{-66} + q^{-68} +2 q^{-70} + q^{-76} }
1,0,0,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -1- q^{-4} + q^{-10} +2 q^{-12} + q^{-14} +3 q^{-16} + q^{-18} +3 q^{-20} + q^{-22} +2 q^{-24} - q^{-30} -2 q^{-32} - q^{-34} -2 q^{-36} - q^{-40} }

B2 Invariants.

Weight	Invariant
0,1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^6+q^4-2 q^2+2-3 q^{-2} +4 q^{-4} -4 q^{-6} +4 q^{-8} -2 q^{-10} +3 q^{-12} + q^{-14} +4 q^{-18} -3 q^{-20} +6 q^{-22} -6 q^{-24} +7 q^{-26} -6 q^{-28} +5 q^{-30} -5 q^{-32} +3 q^{-34} -3 q^{-36} -2 q^{-42} +2 q^{-44} -2 q^{-46} +3 q^{-48} -3 q^{-50} +3 q^{-52} -2 q^{-54} +2 q^{-56} -2 q^{-58} + q^{-60} - q^{-62} }
1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{12}-q^8-q^6+q^4+q^2-2-3 q^{-2} +3 q^{-6} -3 q^{-10} - q^{-12} +4 q^{-14} +4 q^{-16} + q^{-18} -3 q^{-20} + q^{-22} +4 q^{-24} +4 q^{-26} - q^{-28} -2 q^{-30} +3 q^{-34} + q^{-36} - q^{-38} +2 q^{-42} -2 q^{-46} - q^{-48} + q^{-50} + q^{-52} -2 q^{-54} -3 q^{-56} - q^{-58} +2 q^{-60} -2 q^{-64} -2 q^{-66} + q^{-68} +2 q^{-70} -2 q^{-74} -2 q^{-76} + q^{-78} +3 q^{-80} + q^{-82} -2 q^{-84} -2 q^{-86} +2 q^{-90} + q^{-92} - q^{-94} - q^{-96} + q^{-100} }

D4 Invariants.

Weight

Invariant

1,0,0,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^6-q^4+q^2-2+2 q^{-2} -4 q^{-4} +2 q^{-6} -4 q^{-8} +3 q^{-10} -3 q^{-12} +2 q^{-14} - q^{-16} +3 q^{-18} +2 q^{-20} + q^{-22} +5 q^{-24} +2 q^{-26} +8 q^{-28} +8 q^{-32} -3 q^{-34} +6 q^{-36} -6 q^{-38} +2 q^{-40} -7 q^{-42} -6 q^{-46} -3 q^{-50} - q^{-56} + q^{-58} - q^{-60} +2 q^{-62} -2 q^{-64} + q^{-66} -2 q^{-68} +3 q^{-70} -2 q^{-72} + q^{-74} -2 q^{-76} +2 q^{-78} - q^{-80} + q^{-82} - q^{-84} + q^{-86} }

G2 Invariants.

Weight

Invariant

1,0

$q^{12}-q^{10}+2q^{8}-3q^{6}+q^{4}-2q^{2}-1+5q^{-2}-8q^{-4}+8q^{-6}-7q^{-8}+2q^{-10}+3q^{-12}-9q^{-14}+10q^{-16}-9q^{-18}+5q^{-20}+2q^{-22}-4q^{-24}+7q^{-26}-3q^{-28}+2q^{-30}+4q^{-32}-2q^{-34}+2q^{-36}+2q^{-38}-2q^{-40}+8q^{-42}-4q^{-44}+6q^{-46}-2q^{-48}-q^{-50}+7q^{-52}-10q^{-54}+9q^{-56}-6q^{-58}+2q^{-60}+2q^{-62}-6q^{-64}+6q^{-66}-5q^{-68}+2q^{-70}-3q^{-74}+q^{-76}+q^{-78}-2q^{-80}+q^{-82}-q^{-86}+2q^{-88}-2q^{-90}+q^{-92}+q^{-96}-q^{-98}-q^{-100}-q^{-104}+2q^{-106}-4q^{-108}+4q^{-110}-3q^{-112}-q^{-114}+2q^{-116}-5q^{-118}+5q^{-120}-5q^{-122}+3q^{-124}-q^{-126}-2q^{-128}+4q^{-130}-4q^{-132}+4q^{-134}-2q^{-136}+q^{-138}-2q^{-142}+2q^{-144}-q^{-146}+q^{-148}$

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["10 5"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^4-3 t^3+5 t^2-5 t+5-5 t^{-1} +5 t^{-2} -3 t^{-3} + t^{-4} }

In[5]:=

Conway[K][z]

Out[5]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^8+5 z^6+7 z^4+4 z^2+1}

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 33, 4 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^9+2 q^8-3 q^7+4 q^6-5 q^5+5 q^4-4 q^3+4 q^2-2 q+2- q^{-1} }

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^8 a^{-4} -z^6 a^{-2} +7 z^6 a^{-4} -z^6 a^{-6} -5 z^4 a^{-2} +17 z^4 a^{-4} -5 z^4 a^{-6} -6 z^2 a^{-2} +17 z^2 a^{-4} -7 z^2 a^{-6} - a^{-2} +5 a^{-4} -3 a^{-6} }

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^9 a^{-3} +z^9 a^{-5} +2 z^8 a^{-2} +4 z^8 a^{-4} +2 z^8 a^{-6} +z^7 a^{-1} -3 z^7 a^{-3} -2 z^7 a^{-5} +2 z^7 a^{-7} -11 z^6 a^{-2} -20 z^6 a^{-4} -7 z^6 a^{-6} +2 z^6 a^{-8} -5 z^5 a^{-1} -2 z^5 a^{-3} -3 z^5 a^{-5} -4 z^5 a^{-7} +2 z^5 a^{-9} +18 z^4 a^{-2} +32 z^4 a^{-4} +10 z^4 a^{-6} -2 z^4 a^{-8} +2 z^4 a^{-10} +6 z^3 a^{-1} +7 z^3 a^{-3} +6 z^3 a^{-5} +3 z^3 a^{-7} -z^3 a^{-9} +z^3 a^{-11} -10 z^2 a^{-2} -22 z^2 a^{-4} -9 z^2 a^{-6} +z^2 a^{-8} -2 z^2 a^{-10} -z a^{-1} -2 z a^{-3} -3 z a^{-5} -z a^{-7} -z a^{-11} + a^{-2} +5 a^{-4} +3 a^{-6} }

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q\leftrightarrow q^{-1}} ): {}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["10 5"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]=

{ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^4-3 t^3+5 t^2-5 t+5-5 t^{-1} +5 t^{-2} -3 t^{-3} + t^{-4} } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^9+2 q^8-3 q^7+4 q^6-5 q^5+5 q^4-4 q^3+4 q^2-2 q+2- q^{-1} } }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{}

Vassiliev invariants

V2 and V3:

(4, 7)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 16} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 56} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 128} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{728}{3}} ${\frac {16}{3}}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 896} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{3824}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{608}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 24} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{2048}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1568} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{11648}{3}} ${\frac {256}{3}}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{104222}{15}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{18512}{15}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{38288}{45}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{130}{9}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{658}{15}}

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s-1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} 4 is the signature of 10 5. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 5 6 7 χ 19                     1 -1 17                   1   1 15                 2 1   -1 13               2 1     1 11             3 2       -1 9           2 2         0 7         2 3           1 5       2 2             0 3     1 3               2 1   1 1                 0 -1   1                   1 -3 1                     -1

Integral Khovanov Homology (db, data source)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}} $i=3$ $i=5$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-3} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=0} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=3} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=4} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=5} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=6} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=7} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n+1} -dimensional representation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sl(2)} )

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_n}
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{25}-2 q^{24}+4 q^{22}-5 q^{21}+7 q^{19}-8 q^{18}+q^{17}+8 q^{16}-11 q^{15}+3 q^{14}+9 q^{13}-13 q^{12}+3 q^{11}+11 q^{10}-14 q^9+q^8+13 q^7-12 q^6-q^5+13 q^4-9 q^3-3 q^2+10 q-4-4 q^{-1} +5 q^{-2} - q^{-3} -2 q^{-4} + q^{-5} }
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{48}+2 q^{47}-q^{45}-3 q^{44}+4 q^{43}+3 q^{42}-4 q^{41}-7 q^{40}+6 q^{39}+10 q^{38}-6 q^{37}-14 q^{36}+5 q^{35}+18 q^{34}-2 q^{33}-22 q^{32}-q^{31}+24 q^{30}+5 q^{29}-24 q^{28}-9 q^{27}+23 q^{26}+11 q^{25}-22 q^{24}-10 q^{23}+20 q^{22}+7 q^{21}-17 q^{20}-6 q^{19}+19 q^{18}-2 q^{17}-14 q^{16}+18 q^{14}-9 q^{13}-11 q^{12}+6 q^{11}+16 q^{10}-13 q^9-11 q^8+10 q^7+17 q^6-13 q^5-14 q^4+8 q^3+18 q^2-7 q-15+2 q^{-1} +14 q^{-2} + q^{-3} -11 q^{-4} -3 q^{-5} +7 q^{-6} +3 q^{-7} -4 q^{-8} -2 q^{-9} + q^{-10} +2 q^{-11} - q^{-12} }
4	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{78}-2 q^{77}+q^{75}+4 q^{73}-7 q^{72}+q^{71}+2 q^{70}+q^{69}+9 q^{68}-18 q^{67}+q^{66}+5 q^{65}+8 q^{64}+16 q^{63}-36 q^{62}-4 q^{61}+8 q^{60}+25 q^{59}+29 q^{58}-61 q^{57}-19 q^{56}+9 q^{55}+52 q^{54}+49 q^{53}-88 q^{52}-44 q^{51}+4 q^{50}+84 q^{49}+77 q^{48}-107 q^{47}-71 q^{46}-11 q^{45}+104 q^{44}+105 q^{43}-108 q^{42}-86 q^{41}-30 q^{40}+105 q^{39}+118 q^{38}-99 q^{37}-82 q^{36}-37 q^{35}+91 q^{34}+115 q^{33}-89 q^{32}-71 q^{31}-35 q^{30}+75 q^{29}+108 q^{28}-80 q^{27}-58 q^{26}-32 q^{25}+54 q^{24}+99 q^{23}-66 q^{22}-42 q^{21}-29 q^{20}+30 q^{19}+85 q^{18}-52 q^{17}-23 q^{16}-20 q^{15}+11 q^{14}+66 q^{13}-43 q^{12}-10 q^{11}-9 q^{10}+4 q^9+51 q^8-40 q^7-8 q^6-4 q^5+6 q^4+46 q^3-34 q^2-13 q-9+5 q^{-1} +45 q^{-2} -19 q^{-3} -12 q^{-4} -16 q^{-5} -4 q^{-6} +35 q^{-7} -3 q^{-8} -4 q^{-9} -14 q^{-10} -10 q^{-11} +18 q^{-12} +2 q^{-13} +2 q^{-14} -5 q^{-15} -7 q^{-16} +5 q^{-17} + q^{-18} +2 q^{-19} - q^{-20} -2 q^{-21} + q^{-22} }
5	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{115}+2 q^{114}-q^{112}-q^{110}-q^{109}+3 q^{108}+q^{107}-3 q^{106}+q^{105}-q^{104}+5 q^{102}-q^{101}-7 q^{100}-q^{99}+q^{98}+6 q^{97}+11 q^{96}-2 q^{95}-18 q^{94}-13 q^{93}+3 q^{92}+21 q^{91}+26 q^{90}-35 q^{88}-37 q^{87}+2 q^{86}+45 q^{85}+50 q^{84}-3 q^{83}-61 q^{82}-63 q^{81}+7 q^{80}+83 q^{79}+77 q^{78}-17 q^{77}-108 q^{76}-92 q^{75}+27 q^{74}+137 q^{73}+114 q^{72}-37 q^{71}-170 q^{70}-138 q^{69}+41 q^{68}+199 q^{67}+164 q^{66}-37 q^{65}-219 q^{64}-192 q^{63}+26 q^{62}+233 q^{61}+211 q^{60}-15 q^{59}-228 q^{58}-224 q^{57}-2 q^{56}+224 q^{55}+226 q^{54}+12 q^{53}-209 q^{52}-224 q^{51}-23 q^{50}+198 q^{49}+217 q^{48}+33 q^{47}-180 q^{46}-220 q^{45}-43 q^{44}+173 q^{43}+208 q^{42}+58 q^{41}-146 q^{40}-221 q^{39}-70 q^{38}+139 q^{37}+202 q^{36}+90 q^{35}-103 q^{34}-211 q^{33}-103 q^{32}+90 q^{31}+182 q^{30}+117 q^{29}-48 q^{28}-179 q^{27}-122 q^{26}+31 q^{25}+141 q^{24}+124 q^{23}+4 q^{22}-124 q^{21}-114 q^{20}-16 q^{19}+82 q^{18}+104 q^{17}+37 q^{16}-66 q^{15}-82 q^{14}-33 q^{13}+33 q^{12}+66 q^{11}+43 q^{10}-30 q^9-52 q^8-25 q^7+13 q^6+40 q^5+34 q^4-18 q^3-41 q^2-22 q+9+37 q^{-1} +34 q^{-2} -9 q^{-3} -38 q^{-4} -32 q^{-5} -4 q^{-6} +31 q^{-7} +41 q^{-8} +10 q^{-9} -24 q^{-10} -34 q^{-11} -21 q^{-12} +10 q^{-13} +31 q^{-14} +25 q^{-15} -4 q^{-16} -20 q^{-17} -20 q^{-18} -7 q^{-19} +11 q^{-20} +18 q^{-21} +7 q^{-22} -6 q^{-23} -8 q^{-24} -6 q^{-25} -2 q^{-26} +7 q^{-27} +5 q^{-28} - q^{-29} -2 q^{-30} - q^{-31} -2 q^{-32} + q^{-33} +2 q^{-34} - q^{-35} }
6	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{159}-2 q^{158}+q^{156}+q^{154}-2 q^{153}+5 q^{152}-5 q^{151}+q^{149}-2 q^{148}+2 q^{147}-6 q^{146}+13 q^{145}-8 q^{144}+5 q^{143}+q^{142}-7 q^{141}+q^{140}-16 q^{139}+20 q^{138}-11 q^{137}+19 q^{136}+7 q^{135}-7 q^{134}-q^{133}-39 q^{132}+14 q^{131}-24 q^{130}+44 q^{129}+27 q^{128}+9 q^{127}+10 q^{126}-72 q^{125}-12 q^{124}-59 q^{123}+62 q^{122}+51 q^{121}+42 q^{120}+45 q^{119}-89 q^{118}-35 q^{117}-110 q^{116}+56 q^{115}+42 q^{114}+59 q^{113}+94 q^{112}-65 q^{111}-6 q^{110}-134 q^{109}+32 q^{108}-33 q^{107}+5 q^{106}+112 q^{105}-7 q^{104}+104 q^{103}-76 q^{102}+37 q^{101}-150 q^{100}-136 q^{99}+46 q^{98}+23 q^{97}+247 q^{96}+65 q^{95}+125 q^{94}-225 q^{93}-291 q^{92}-92 q^{91}-35 q^{90}+323 q^{89}+203 q^{88}+270 q^{87}-201 q^{86}-362 q^{85}-209 q^{84}-139 q^{83}+295 q^{82}+251 q^{81}+372 q^{80}-133 q^{79}-334 q^{78}-235 q^{77}-200 q^{76}+231 q^{75}+216 q^{74}+387 q^{73}-87 q^{72}-274 q^{71}-203 q^{70}-210 q^{69}+180 q^{68}+162 q^{67}+365 q^{66}-50 q^{65}-214 q^{64}-172 q^{63}-225 q^{62}+122 q^{61}+107 q^{60}+356 q^{59}+22 q^{58}-135 q^{57}-153 q^{56}-270 q^{55}+32 q^{54}+35 q^{53}+353 q^{52}+128 q^{51}-20 q^{50}-120 q^{49}-323 q^{48}-84 q^{47}-66 q^{46}+328 q^{45}+233 q^{44}+117 q^{43}-53 q^{42}-342 q^{41}-191 q^{40}-192 q^{39}+252 q^{38}+290 q^{37}+242 q^{36}+54 q^{35}-290 q^{34}-242 q^{33}-310 q^{32}+119 q^{31}+257 q^{30}+304 q^{29}+167 q^{28}-164 q^{27}-196 q^{26}-365 q^{25}-25 q^{24}+140 q^{23}+265 q^{22}+223 q^{21}-24 q^{20}-71 q^{19}-319 q^{18}-109 q^{17}+12 q^{16}+152 q^{15}+186 q^{14}+55 q^{13}+52 q^{12}-212 q^{11}-103 q^{10}-53 q^9+50 q^8+102 q^7+57 q^6+106 q^5-127 q^4-60 q^3-54 q^2+7 q+45+40 q^{-1} +109 q^{-2} -89 q^{-3} -41 q^{-4} -51 q^{-5} -7 q^{-6} +22 q^{-7} +43 q^{-8} +113 q^{-9} -57 q^{-10} -32 q^{-11} -60 q^{-12} -32 q^{-13} -8 q^{-14} +36 q^{-15} +115 q^{-16} -11 q^{-17} -4 q^{-18} -47 q^{-19} -44 q^{-20} -42 q^{-21} +3 q^{-22} +83 q^{-23} +15 q^{-24} +24 q^{-25} -14 q^{-26} -24 q^{-27} -44 q^{-28} -21 q^{-29} +37 q^{-30} +8 q^{-31} +23 q^{-32} +6 q^{-33} - q^{-34} -22 q^{-35} -18 q^{-36} +10 q^{-37} - q^{-38} +9 q^{-39} +5 q^{-40} +5 q^{-41} -7 q^{-42} -7 q^{-43} +3 q^{-44} -2 q^{-45} +2 q^{-46} + q^{-47} +2 q^{-48} - q^{-49} -2 q^{-50} + q^{-51} }
7	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{210}+2 q^{209}-q^{207}-q^{205}+2 q^{204}-2 q^{203}-3 q^{202}+4 q^{201}+2 q^{200}+q^{198}-3 q^{197}+5 q^{196}-4 q^{195}-12 q^{194}+4 q^{193}+3 q^{192}+5 q^{191}+7 q^{190}-2 q^{189}+11 q^{188}-5 q^{187}-24 q^{186}-6 q^{185}-8 q^{184}+7 q^{183}+22 q^{182}+11 q^{181}+30 q^{180}+6 q^{179}-36 q^{178}-30 q^{177}-46 q^{176}-18 q^{175}+30 q^{174}+43 q^{173}+84 q^{172}+52 q^{171}-25 q^{170}-56 q^{169}-122 q^{168}-104 q^{167}-10 q^{166}+68 q^{165}+176 q^{164}+167 q^{163}+55 q^{162}-50 q^{161}-217 q^{160}-264 q^{159}-141 q^{158}+22 q^{157}+264 q^{156}+359 q^{155}+246 q^{154}+51 q^{153}-277 q^{152}-469 q^{151}-396 q^{150}-165 q^{149}+269 q^{148}+580 q^{147}+566 q^{146}+321 q^{145}-206 q^{144}-663 q^{143}-765 q^{142}-543 q^{141}+89 q^{140}+723 q^{139}+973 q^{138}+798 q^{137}+92 q^{136}-725 q^{135}-1155 q^{134}-1087 q^{133}-342 q^{132}+656 q^{131}+1304 q^{130}+1377 q^{129}+623 q^{128}-520 q^{127}-1371 q^{126}-1616 q^{125}-926 q^{124}+315 q^{123}+1360 q^{122}+1790 q^{121}+1198 q^{120}-89 q^{119}-1271 q^{118}-1871 q^{117}-1398 q^{116}-129 q^{115}+1129 q^{114}+1860 q^{113}+1520 q^{112}+311 q^{111}-985 q^{110}-1803 q^{109}-1556 q^{108}-402 q^{107}+862 q^{106}+1699 q^{105}+1528 q^{104}+454 q^{103}-782 q^{102}-1624 q^{101}-1473 q^{100}-435 q^{99}+750 q^{98}+1550 q^{97}+1398 q^{96}+407 q^{95}-728 q^{94}-1497 q^{93}-1349 q^{92}-378 q^{91}+728 q^{90}+1454 q^{89}+1277 q^{88}+356 q^{87}-685 q^{86}-1388 q^{85}-1236 q^{84}-356 q^{83}+647 q^{82}+1310 q^{81}+1150 q^{80}+362 q^{79}-552 q^{78}-1196 q^{77}-1085 q^{76}-371 q^{75}+467 q^{74}+1060 q^{73}+963 q^{72}+380 q^{71}-335 q^{70}-900 q^{69}-855 q^{68}-372 q^{67}+228 q^{66}+719 q^{65}+681 q^{64}+354 q^{63}-91 q^{62}-523 q^{61}-528 q^{60}-301 q^{59}+11 q^{58}+325 q^{57}+304 q^{56}+215 q^{55}+91 q^{54}-140 q^{53}-120 q^{52}-99 q^{51}-87 q^{50}-9 q^{49}-119 q^{48}-59 q^{47}+92 q^{46}+119 q^{45}+272 q^{44}+215 q^{43}+17 q^{42}-145 q^{41}-430 q^{40}-395 q^{39}-109 q^{38}+120 q^{37}+478 q^{36}+506 q^{35}+271 q^{34}-13 q^{33}-480 q^{32}-594 q^{31}-382 q^{30}-115 q^{29}+383 q^{28}+585 q^{27}+483 q^{26}+260 q^{25}-255 q^{24}-518 q^{23}-490 q^{22}-382 q^{21}+89 q^{20}+393 q^{19}+466 q^{18}+433 q^{17}+39 q^{16}-235 q^{15}-347 q^{14}-447 q^{13}-153 q^{12}+104 q^{11}+243 q^{10}+383 q^9+179 q^8+12 q^7-101 q^6-314 q^5-192 q^4-64 q^3+24 q^2+220 q+151+93 q^{-1} +47 q^{-2} -162 q^{-3} -124 q^{-4} -85 q^{-5} -74 q^{-6} +106 q^{-7} +97 q^{-8} +91 q^{-9} +93 q^{-10} -85 q^{-11} -82 q^{-12} -80 q^{-13} -109 q^{-14} +44 q^{-15} +66 q^{-16} +95 q^{-17} +128 q^{-18} -20 q^{-19} -51 q^{-20} -78 q^{-21} -135 q^{-22} -26 q^{-23} +12 q^{-24} +68 q^{-25} +144 q^{-26} +49 q^{-27} +12 q^{-28} -35 q^{-29} -115 q^{-30} -63 q^{-31} -50 q^{-32} - q^{-33} +92 q^{-34} +67 q^{-35} +52 q^{-36} +21 q^{-37} -52 q^{-38} -39 q^{-39} -54 q^{-40} -44 q^{-41} +25 q^{-42} +29 q^{-43} +42 q^{-44} +33 q^{-45} -9 q^{-46} -5 q^{-47} -20 q^{-48} -33 q^{-49} -6 q^{-50} +2 q^{-51} +16 q^{-52} +19 q^{-53} -2 q^{-54} +4 q^{-55} - q^{-56} -11 q^{-57} -5 q^{-58} -4 q^{-59} +4 q^{-60} +7 q^{-61} - q^{-62} +2 q^{-64} -2 q^{-65} - q^{-66} -2 q^{-67} + q^{-68} +2 q^{-69} - q^{-70} }

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Rolfsen Knot Page master template (intermediate). See/edit the Rolfsen_Splice_Base (expert). Back to the top.

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