10 61

10_60

10_62

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 10 61's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 61 at Knotilus!
	10_61 is also known as the pretzel knot P(4,3,3).

Knot presentations

Planar diagram presentation	X8291 X10,4,11,3 X2,10,3,9 X18,12,19,11 X14,7,15,8 X16,5,17,6 X6,15,7,16 X4,17,5,18 X20,14,1,13 X12,20,13,19
Gauss code	1, -3, 2, -8, 6, -7, 5, -1, 3, -2, 4, -10, 9, -5, 7, -6, 8, -4, 10, -9
Dowker-Thistlethwaite code	8 10 16 14 2 18 20 6 4 12
Conway Notation	[4,3,3]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 11, width is 4, Braid index is 4		[{6, 13}, {1, 12}, {13, 11}, {12, 4}, {10, 3}, {11, 9}, {8, 10}, {9, 7}, {5, 8}, {4, 2}, {3, 6}, {2, 5}, {7, 1}]

[edit Notes on presentations of 10 61]

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 61"];

In[4]:=

PD[K]

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

Out[4]=

X8291 X10,4,11,3 X2,10,3,9 X18,12,19,11 X14,7,15,8 X16,5,17,6 X6,15,7,16 X4,17,5,18 X20,14,1,13 X12,20,13,19

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

8 10 16 14 2 18 20 6 4 12

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[4,3,3]

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]=

$\textrm{BR}(4,\{1,1,1,-2,1,1,1,-2,-3,2,-3\})$

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]=

{ 4, 11, 4 }

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[{6, 13}, {1, 12}, {13, 11}, {12, 4}, {10, 3}, {11, 9}, {8, 10}, {9, 7}, {5, 8}, {4, 2}, {3, 6}, {2, 5}, {7, 1}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{2,3\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 2 Maximal Thurston-Bennequin number [-1][-11] Hyperbolic Volume 8.45858 A-Polynomial See Data:10 61/A-polynomial

[edit Notes for 10 61's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $2$ Topological 4 genus $2$ Concordance genus $3$ Rasmussen s-Invariant -4

[edit Notes for 10 61's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2 t^3+5 t^2-6 t+7-6 t^{-1} +5 t^{-2} -2 t^{-3}$
Conway polynomial	$-2 z^6-7 z^4-4 z^2+1$
2nd Alexander ideal (db, data sources)	$\left\{2,t^2+t+1\right\}$
Determinant and Signature	{ 33, 4 }
Jones polynomial	$q^8-2 q^7+3 q^6-4 q^5+5 q^4-5 q^3+4 q^2-4 q+3- q^{-1} + q^{-2}$
HOMFLY-PT polynomial (db, data sources)	$-z^6 a^{-2} -z^6 a^{-4} -5 z^4 a^{-2} -4 z^4 a^{-4} +z^4 a^{-6} +z^4-8 z^2 a^{-2} -3 z^2 a^{-4} +3 z^2 a^{-6} +4 z^2-5 a^{-2} + a^{-4} + a^{-6} +4$
Kauffman polynomial (db, data sources)	$z^9 a^{-1} +z^9 a^{-3} +4 z^8 a^{-2} +3 z^8 a^{-4} +z^8-5 z^7 a^{-1} +5 z^7 a^{-5} -22 z^6 a^{-2} -10 z^6 a^{-4} +5 z^6 a^{-6} -7 z^6+6 z^5 a^{-1} -16 z^5 a^{-3} -18 z^5 a^{-5} +4 z^5 a^{-7} +38 z^4 a^{-2} +5 z^4 a^{-4} -13 z^4 a^{-6} +3 z^4 a^{-8} +17 z^4+z^3 a^{-1} +26 z^3 a^{-3} +17 z^3 a^{-5} -6 z^3 a^{-7} +2 z^3 a^{-9} -24 z^2 a^{-2} +z^2 a^{-4} +6 z^2 a^{-6} -2 z^2 a^{-8} +z^2 a^{-10} -16 z^2-2 z a^{-1} -8 z a^{-3} -6 z a^{-5} +5 a^{-2} + a^{-4} - a^{-6} +4$
The A2 invariant	$q^6+q^4+2 q^2+2- q^{-4} -3 q^{-6} -2 q^{-8} +2 q^{-14} + q^{-24}$
The G2 invariant	$q^{26}+3 q^{22}-3 q^{20}+3 q^{18}-q^{16}+7 q^{12}-9 q^{10}+10 q^8-3 q^6+q^4+8 q^2-12+11 q^{-2} -2 q^{-4} +5 q^{-8} -9 q^{-10} +4 q^{-12} +5 q^{-14} -4 q^{-16} + q^{-18} -5 q^{-20} - q^{-22} +4 q^{-24} -8 q^{-26} +4 q^{-28} -9 q^{-30} +5 q^{-32} + q^{-34} -7 q^{-36} +6 q^{-38} -12 q^{-40} +8 q^{-42} -5 q^{-44} -3 q^{-46} +7 q^{-48} -9 q^{-50} +5 q^{-52} +3 q^{-54} -3 q^{-56} +4 q^{-58} - q^{-60} -4 q^{-62} +6 q^{-64} -3 q^{-66} +3 q^{-68} + q^{-70} -2 q^{-72} +6 q^{-74} -3 q^{-76} +3 q^{-78} - q^{-80} + q^{-82} - q^{-84} + q^{-86} + q^{-88} -2 q^{-90} +4 q^{-92} -3 q^{-94} +2 q^{-96} - q^{-98} - q^{-100} -3 q^{-104} +3 q^{-106} - q^{-108} + q^{-110} - q^{-114} +2 q^{-116} -2 q^{-118} +2 q^{-120} - q^{-122} - q^{-128} + q^{-130} - q^{-132} + q^{-134}$

Further Quantum Invariants

Further quantum knot invariants for 10_61.

A1 Invariants.

Weight	Invariant
1	$q^5+2 q- q^{-1} - q^{-5} + q^{-9} - q^{-11} + q^{-13} - q^{-15} + q^{-17}$
2	$q^{18}-q^{14}+2 q^{12}+2 q^{10}-3 q^8+2 q^4-3 q^2-2+2 q^{-2} - q^{-6} +3 q^{-8} +3 q^{-10} - q^{-12} - q^{-14} +2 q^{-16} - q^{-18} -3 q^{-20} +2 q^{-22} + q^{-24} - q^{-26} + q^{-30} - q^{-32} - q^{-34} + q^{-36} - q^{-38} + q^{-40} - q^{-44} + q^{-46}$
3	$q^{39}-q^{35}-q^{33}+2 q^{31}+3 q^{29}-5 q^{25}-2 q^{23}+4 q^{21}+5 q^{19}-3 q^{17}-7 q^{15}-3 q^{13}+5 q^{11}+5 q^9-2 q^7-5 q^5-q^3+7 q+6 q^{-1} - q^{-3} -5 q^{-5} + q^{-7} +5 q^{-9} +2 q^{-11} -6 q^{-13} -3 q^{-15} +3 q^{-17} +3 q^{-19} -4 q^{-21} -4 q^{-23} +5 q^{-25} +6 q^{-27} -5 q^{-29} -8 q^{-31} +3 q^{-33} +8 q^{-35} -8 q^{-39} -4 q^{-41} +3 q^{-43} +7 q^{-45} +3 q^{-47} -6 q^{-49} -6 q^{-51} +6 q^{-53} +9 q^{-55} -3 q^{-57} -8 q^{-59} +5 q^{-63} -3 q^{-67} + q^{-69} + q^{-71} -2 q^{-75} + q^{-79} - q^{-85} + q^{-87}$
4	$q^{68}-q^{64}-q^{62}-q^{60}+3 q^{58}+3 q^{56}+q^{54}-2 q^{52}-8 q^{50}-2 q^{48}+4 q^{46}+9 q^{44}+7 q^{42}-7 q^{40}-11 q^{38}-11 q^{36}+2 q^{34}+16 q^{32}+11 q^{30}+2 q^{28}-16 q^{26}-18 q^{24}-q^{22}+13 q^{20}+25 q^{18}+9 q^{16}-14 q^{14}-21 q^{12}-14 q^{10}+15 q^8+28 q^6+12 q^4-11 q^2-31-17 q^{-2} +13 q^{-4} +26 q^{-6} +17 q^{-8} -15 q^{-10} -29 q^{-12} -12 q^{-14} +14 q^{-16} +29 q^{-18} +12 q^{-20} -17 q^{-22} -22 q^{-24} -5 q^{-26} +21 q^{-28} +17 q^{-30} -7 q^{-32} -17 q^{-34} -9 q^{-36} +13 q^{-38} +13 q^{-40} -9 q^{-42} -16 q^{-44} -2 q^{-46} +22 q^{-48} +20 q^{-50} -14 q^{-52} -28 q^{-54} -12 q^{-56} +21 q^{-58} +34 q^{-60} +6 q^{-62} -25 q^{-64} -32 q^{-66} -10 q^{-68} +25 q^{-70} +33 q^{-72} +11 q^{-74} -21 q^{-76} -40 q^{-78} -12 q^{-80} +26 q^{-82} +37 q^{-84} +11 q^{-86} -33 q^{-88} -32 q^{-90} +2 q^{-92} +29 q^{-94} +22 q^{-96} -14 q^{-98} -21 q^{-100} -4 q^{-102} +12 q^{-104} +14 q^{-106} -4 q^{-108} -7 q^{-110} -3 q^{-112} +2 q^{-114} +4 q^{-116} -4 q^{-118} + q^{-120} + q^{-122} -3 q^{-128} +2 q^{-130} - q^{-138} + q^{-140}$
5	$q^{105}-q^{101}-q^{99}-q^{97}+3 q^{93}+4 q^{91}+q^{89}-2 q^{87}-5 q^{85}-8 q^{83}-3 q^{81}+6 q^{79}+12 q^{77}+11 q^{75}+3 q^{73}-11 q^{71}-20 q^{69}-17 q^{67}-2 q^{65}+17 q^{63}+28 q^{61}+22 q^{59}+q^{57}-25 q^{55}-39 q^{53}-27 q^{51}+3 q^{49}+34 q^{47}+49 q^{45}+35 q^{43}-7 q^{41}-46 q^{39}-57 q^{37}-33 q^{35}+14 q^{33}+61 q^{31}+70 q^{29}+30 q^{27}-31 q^{25}-74 q^{23}-75 q^{21}-23 q^{19}+49 q^{17}+88 q^{15}+67 q^{13}+3 q^{11}-70 q^9-99 q^7-58 q^5+24 q^3+92 q+97 q^{-1} +35 q^{-3} -55 q^{-5} -106 q^{-7} -81 q^{-9} +5 q^{-11} +91 q^{-13} +114 q^{-15} +49 q^{-17} -51 q^{-19} -116 q^{-21} -91 q^{-23} +5 q^{-25} +99 q^{-27} +112 q^{-29} +38 q^{-31} -66 q^{-33} -116 q^{-35} -73 q^{-37} +30 q^{-39} +104 q^{-41} +88 q^{-43} +5 q^{-45} -78 q^{-47} -90 q^{-49} -24 q^{-51} +51 q^{-53} +71 q^{-55} +28 q^{-57} -33 q^{-59} -48 q^{-61} -16 q^{-63} +29 q^{-65} +31 q^{-67} -12 q^{-69} -45 q^{-71} -26 q^{-73} +29 q^{-75} +73 q^{-77} +49 q^{-79} -33 q^{-81} -99 q^{-83} -81 q^{-85} +9 q^{-87} +99 q^{-89} +117 q^{-91} +43 q^{-93} -66 q^{-95} -127 q^{-97} -94 q^{-99} +2 q^{-101} +96 q^{-103} +127 q^{-105} +74 q^{-107} -37 q^{-109} -129 q^{-111} -128 q^{-113} -35 q^{-115} +93 q^{-117} +155 q^{-119} +94 q^{-121} -42 q^{-123} -145 q^{-125} -124 q^{-127} -2 q^{-129} +110 q^{-131} +125 q^{-133} +32 q^{-135} -78 q^{-137} -108 q^{-139} -41 q^{-141} +51 q^{-143} +82 q^{-145} +38 q^{-147} -32 q^{-149} -61 q^{-151} -28 q^{-153} +24 q^{-155} +39 q^{-157} +16 q^{-159} -15 q^{-161} -25 q^{-163} -9 q^{-165} +12 q^{-167} +15 q^{-169} +3 q^{-171} -6 q^{-173} -5 q^{-175} - q^{-177} + q^{-179} +2 q^{-181} - q^{-185} + q^{-187} - q^{-191} - q^{-193} + q^{-195} - q^{-203} + q^{-205}$

A2 Invariants.

Weight	Invariant
1,0	$q^6+q^4+2 q^2+2- q^{-4} -3 q^{-6} -2 q^{-8} +2 q^{-14} + q^{-24}$
1,1	$q^{20}+6 q^{16}-6 q^{14}+14 q^{12}-18 q^{10}+26 q^8-24 q^6+22 q^4-22 q^2+6-15 q^{-4} +16 q^{-6} -26 q^{-8} +34 q^{-10} -32 q^{-12} +34 q^{-14} -28 q^{-16} +28 q^{-18} -13 q^{-20} +12 q^{-22} +2 q^{-24} -4 q^{-26} +6 q^{-28} -8 q^{-30} -6 q^{-34} +3 q^{-36} -2 q^{-38} +4 q^{-40} -8 q^{-42} +7 q^{-44} -6 q^{-46} +6 q^{-48} -6 q^{-50} +5 q^{-52} -2 q^{-54} +4 q^{-56} -4 q^{-58} +3 q^{-60} -2 q^{-62} +2 q^{-64} -2 q^{-66} + q^{-68}$
2,0	$q^{20}+q^{18}+q^{16}+q^{14}+3 q^{12}+3 q^{10}+2 q^8-q^6-q^4-3 q^2-5-7 q^{-2} -6 q^{-4} -2 q^{-6} +3 q^{-10} +7 q^{-12} +9 q^{-14} +6 q^{-16} +4 q^{-18} -2 q^{-22} -4 q^{-24} -3 q^{-26} -2 q^{-28} -2 q^{-30} +2 q^{-32} +4 q^{-34} + q^{-36} -2 q^{-38} - q^{-40} -2 q^{-42} -2 q^{-44} - q^{-46} + q^{-48} +2 q^{-50} + q^{-60}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{12}+3 q^8+2 q^6+2 q^4+3 q^2+1- q^{-2} -2 q^{-6} -4 q^{-8} -2 q^{-10} -4 q^{-12} -2 q^{-14} +2 q^{-20} +4 q^{-22} +3 q^{-24} + q^{-26} - q^{-32} -2 q^{-34} + q^{-36} + q^{-38} - q^{-40} + q^{-42} + q^{-44} -2 q^{-46} +2 q^{-50} - q^{-52} - q^{-54} + q^{-56}$
1,0,0	$q^7+q^5+3 q^3+2 q+3 q^{-1} - q^{-5} -4 q^{-7} -3 q^{-9} -3 q^{-11} - q^{-13} + q^{-15} + q^{-17} +2 q^{-19} + q^{-23} - q^{-25} + q^{-27} + q^{-31}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{14}+q^{12}+3 q^{10}+5 q^8+6 q^6+6 q^4+7 q^2+2- q^{-2} -5 q^{-4} -9 q^{-6} -12 q^{-8} -10 q^{-10} -6 q^{-12} -5 q^{-14} +6 q^{-18} +8 q^{-20} +2 q^{-22} +6 q^{-24} +6 q^{-26} + q^{-28} + q^{-30} +2 q^{-32} + q^{-34} -2 q^{-36} - q^{-38} -2 q^{-40} -3 q^{-42} -3 q^{-44} + q^{-46} + q^{-48} - q^{-50} +2 q^{-52} +2 q^{-54} - q^{-58} + q^{-62} - q^{-66} + q^{-70}$
1,0,0,0	$q^8+q^6+3 q^4+3 q^2+3+3 q^{-2} - q^{-6} -4 q^{-8} -4 q^{-10} -4 q^{-12} -3 q^{-14} - q^{-16} +2 q^{-20} + q^{-22} +2 q^{-24} + q^{-28} + q^{-34} + q^{-38}$

B2 Invariants.

Weight	Invariant
0,1	$q^{12}+3 q^8-2 q^6+4 q^4-3 q^2+5-3 q^{-2} +4 q^{-4} -2 q^{-6} -4 q^{-12} +4 q^{-14} -8 q^{-16} +6 q^{-18} -8 q^{-20} +6 q^{-22} -7 q^{-24} +5 q^{-26} -2 q^{-28} +2 q^{-30} + q^{-32} +3 q^{-36} -3 q^{-38} +3 q^{-40} -3 q^{-42} +3 q^{-44} -2 q^{-46} +2 q^{-48} -2 q^{-50} + q^{-52} - q^{-54} + q^{-56}$
1,0	$q^{22}+3 q^{14}+2 q^{12}-q^{10}-q^8+3 q^6+3 q^4-4- q^{-2} +2 q^{-4} +2 q^{-6} -3 q^{-8} -5 q^{-10} +3 q^{-14} + q^{-16} -4 q^{-18} -3 q^{-20} + q^{-22} +3 q^{-24} - q^{-26} -2 q^{-28} +3 q^{-32} - q^{-36} + q^{-38} +4 q^{-40} +2 q^{-42} -2 q^{-44} -2 q^{-46} + q^{-48} +3 q^{-50} - q^{-52} -3 q^{-54} -2 q^{-56} +2 q^{-58} +2 q^{-60} - q^{-62} -2 q^{-64} +2 q^{-68} +2 q^{-70} - q^{-72} -2 q^{-74} - q^{-76} + q^{-78} +2 q^{-80} - q^{-84} - q^{-86} + q^{-90}$

D4 Invariants.

Weight

Invariant

1,0,0,0

$q^{14}+3 q^{10}+6 q^6+6 q^2+6 q^{-2} -2 q^{-4} +2 q^{-6} -4 q^{-8} -2 q^{-10} -5 q^{-12} -7 q^{-14} -3 q^{-16} -7 q^{-18} + q^{-20} -7 q^{-22} +6 q^{-24} -3 q^{-26} +10 q^{-28} -2 q^{-30} +8 q^{-32} -2 q^{-34} +5 q^{-36} -2 q^{-38} + q^{-40} -2 q^{-42} - q^{-44} + q^{-46} - q^{-48} +2 q^{-50} -2 q^{-52} +3 q^{-54} -2 q^{-56} +2 q^{-58} -2 q^{-60} +2 q^{-62} -2 q^{-64} + q^{-66} - q^{-68} +2 q^{-70} - q^{-72} - q^{-76} + q^{-78}$

G2 Invariants.

Weight

Invariant

1,0

$q^{26}+3 q^{22}-3 q^{20}+3 q^{18}-q^{16}+7 q^{12}-9 q^{10}+10 q^8-3 q^6+q^4+8 q^2-12+11 q^{-2} -2 q^{-4} +5 q^{-8} -9 q^{-10} +4 q^{-12} +5 q^{-14} -4 q^{-16} + q^{-18} -5 q^{-20} - q^{-22} +4 q^{-24} -8 q^{-26} +4 q^{-28} -9 q^{-30} +5 q^{-32} + q^{-34} -7 q^{-36} +6 q^{-38} -12 q^{-40} +8 q^{-42} -5 q^{-44} -3 q^{-46} +7 q^{-48} -9 q^{-50} +5 q^{-52} +3 q^{-54} -3 q^{-56} +4 q^{-58} - q^{-60} -4 q^{-62} +6 q^{-64} -3 q^{-66} +3 q^{-68} + q^{-70} -2 q^{-72} +6 q^{-74} -3 q^{-76} +3 q^{-78} - q^{-80} + q^{-82} - q^{-84} + q^{-86} + q^{-88} -2 q^{-90} +4 q^{-92} -3 q^{-94} +2 q^{-96} - q^{-98} - q^{-100} -3 q^{-104} +3 q^{-106} - q^{-108} + q^{-110} - q^{-114} +2 q^{-116} -2 q^{-118} +2 q^{-120} - q^{-122} - q^{-128} + q^{-130} - q^{-132} + q^{-134}$

. </div></div>

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 61"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2 t^3+5 t^2-6 t+7-6 t^{-1} +5 t^{-2} -2 t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$-2 z^6-7 z^4-4 z^2+1$

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]=

$\left\{2,t^2+t+1\right\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 33, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^8-2 q^7+3 q^6-4 q^5+5 q^4-5 q^3+4 q^2-4 q+3- q^{-1} + q^{-2}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^6 a^{-2} -z^6 a^{-4} -5 z^4 a^{-2} -4 z^4 a^{-4} +z^4 a^{-6} +z^4-8 z^2 a^{-2} -3 z^2 a^{-4} +3 z^2 a^{-6} +4 z^2-5 a^{-2} + a^{-4} + a^{-6} +4$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^9 a^{-1} +z^9 a^{-3} +4 z^8 a^{-2} +3 z^8 a^{-4} +z^8-5 z^7 a^{-1} +5 z^7 a^{-5} -22 z^6 a^{-2} -10 z^6 a^{-4} +5 z^6 a^{-6} -7 z^6+6 z^5 a^{-1} -16 z^5 a^{-3} -18 z^5 a^{-5} +4 z^5 a^{-7} +38 z^4 a^{-2} +5 z^4 a^{-4} -13 z^4 a^{-6} +3 z^4 a^{-8} +17 z^4+z^3 a^{-1} +26 z^3 a^{-3} +17 z^3 a^{-5} -6 z^3 a^{-7} +2 z^3 a^{-9} -24 z^2 a^{-2} +z^2 a^{-4} +6 z^2 a^{-6} -2 z^2 a^{-8} +z^2 a^{-10} -16 z^2-2 z a^{-1} -8 z a^{-3} -6 z a^{-5} +5 a^{-2} + a^{-4} - a^{-6} +4$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {}

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 61"];

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]=

{ $-2 t^3+5 t^2-6 t+7-6 t^{-1} +5 t^{-2} -2 t^{-3}$, $q^8-2 q^7+3 q^6-4 q^5+5 q^4-5 q^3+4 q^2-4 q+3- q^{-1} + q^{-2}$ }

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`.

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Vassiliev invariants

V2 and V3:

(-4, -5)

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 $-16$ $-40$ $128$ $\frac{568}{3}$ $\frac{368}{3}$ $640$ $\frac{3536}{3}$ $\frac{1088}{3}$ $248$ $-\frac{2048}{3}$ $800$ $-\frac{9088}{3}$ $-\frac{5888}{3}$ $-\frac{8822}{15}$ $\frac{28208}{15}$ $-\frac{156128}{45}$ $\frac{5894}{9}$ $-\frac{16022}{15}$

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 $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$4 is the signature of 10 61. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=3$ $i=5$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the $n+1$-dimensional representation of $sl(2)$)

<p>

$n$	$J_n$
2	$q^{22}-2 q^{21}+q^{20}+2 q^{19}-4 q^{18}+3 q^{17}-4 q^{15}+5 q^{14}-q^{13}-5 q^{12}+7 q^{11}-10 q^9+9 q^8+3 q^7-13 q^6+9 q^5+7 q^4-13 q^3+5 q^2+8 q-11+ q^{-1} +7 q^{-2} -6 q^{-3} - q^{-4} +4 q^{-5} - q^{-6} - q^{-7} + q^{-8}$
3	$q^{42}-2 q^{41}+q^{40}+2 q^{38}-3 q^{37}-q^{36}+2 q^{35}+3 q^{34}-3 q^{33}-5 q^{32}+5 q^{31}+8 q^{30}-8 q^{29}-13 q^{28}+10 q^{27}+20 q^{26}-11 q^{25}-25 q^{24}+10 q^{23}+29 q^{22}-7 q^{21}-29 q^{20}+3 q^{19}+25 q^{18}+q^{17}-21 q^{16}-2 q^{15}+14 q^{14}+4 q^{13}-10 q^{12}-3 q^{11}+5 q^{10}+4 q^9-3 q^8-3 q^7-q^6+q^5+5 q^4-5 q^2-5 q+9+7 q^{-1} -4 q^{-2} -13 q^{-3} +5 q^{-4} +10 q^{-5} +3 q^{-6} -13 q^{-7} -3 q^{-8} +6 q^{-9} +7 q^{-10} -5 q^{-11} -4 q^{-12} +4 q^{-14} - q^{-16} - q^{-17} + q^{-18}$
4	$q^{68}-2 q^{67}+q^{66}+3 q^{63}-7 q^{62}+4 q^{61}+q^{59}+3 q^{58}-12 q^{57}+12 q^{56}-2 q^{55}-4 q^{54}-q^{53}-9 q^{52}+30 q^{51}-4 q^{50}-20 q^{49}-18 q^{48}-2 q^{47}+66 q^{46}+3 q^{45}-47 q^{44}-52 q^{43}-3 q^{42}+110 q^{41}+29 q^{40}-58 q^{39}-90 q^{38}-31 q^{37}+129 q^{36}+61 q^{35}-36 q^{34}-98 q^{33}-66 q^{32}+107 q^{31}+68 q^{30}-5 q^{29}-70 q^{28}-79 q^{27}+74 q^{26}+52 q^{25}+9 q^{24}-36 q^{23}-77 q^{22}+50 q^{21}+38 q^{20}+16 q^{19}-14 q^{18}-77 q^{17}+28 q^{16}+30 q^{15}+26 q^{14}+10 q^{13}-73 q^{12}+2 q^{11}+13 q^{10}+31 q^9+39 q^8-56 q^7-13 q^6-13 q^5+14 q^4+53 q^3-24 q^2-4 q-26-16 q^{-1} +39 q^{-2} -4 q^{-3} +19 q^{-4} -10 q^{-5} -29 q^{-6} +10 q^{-7} -11 q^{-8} +26 q^{-9} +13 q^{-10} -13 q^{-11} -2 q^{-12} -25 q^{-13} +9 q^{-14} +15 q^{-15} +5 q^{-16} +7 q^{-17} -20 q^{-18} -5 q^{-19} +2 q^{-20} +5 q^{-21} +11 q^{-22} -6 q^{-23} -3 q^{-24} -3 q^{-25} - q^{-26} +5 q^{-27} - q^{-30} - q^{-31} + q^{-32}$
5	$q^{100}-2 q^{99}+q^{98}+q^{95}-q^{94}-2 q^{93}+2 q^{92}+q^{91}-2 q^{90}+2 q^{89}+q^{88}-3 q^{87}-3 q^{85}-3 q^{84}+11 q^{83}+13 q^{82}-6 q^{81}-21 q^{80}-19 q^{79}+7 q^{78}+42 q^{77}+36 q^{76}-21 q^{75}-73 q^{74}-52 q^{73}+36 q^{72}+112 q^{71}+80 q^{70}-52 q^{69}-165 q^{68}-119 q^{67}+66 q^{66}+222 q^{65}+173 q^{64}-67 q^{63}-277 q^{62}-241 q^{61}+45 q^{60}+325 q^{59}+309 q^{58}-6 q^{57}-339 q^{56}-369 q^{55}-48 q^{54}+324 q^{53}+401 q^{52}+105 q^{51}-286 q^{50}-400 q^{49}-142 q^{48}+228 q^{47}+368 q^{46}+166 q^{45}-177 q^{44}-326 q^{43}-160 q^{42}+138 q^{41}+278 q^{40}+148 q^{39}-111 q^{38}-244 q^{37}-136 q^{36}+94 q^{35}+223 q^{34}+129 q^{33}-78 q^{32}-201 q^{31}-138 q^{30}+49 q^{29}+191 q^{28}+144 q^{27}-17 q^{26}-158 q^{25}-158 q^{24}-26 q^{23}+125 q^{22}+156 q^{21}+66 q^{20}-75 q^{19}-142 q^{18}-100 q^{17}+22 q^{16}+113 q^{15}+116 q^{14}+29 q^{13}-68 q^{12}-113 q^{11}-72 q^{10}+17 q^9+91 q^8+94 q^7+32 q^6-48 q^5-95 q^4-69 q^3+5 q^2+69 q+83+42 q^{-1} -33 q^{-2} -74 q^{-3} -63 q^{-4} -13 q^{-5} +42 q^{-6} +71 q^{-7} +40 q^{-8} -10 q^{-9} -42 q^{-10} -52 q^{-11} -30 q^{-12} +19 q^{-13} +41 q^{-14} +33 q^{-15} +19 q^{-16} -12 q^{-17} -39 q^{-18} -28 q^{-19} -6 q^{-20} +9 q^{-21} +30 q^{-22} +27 q^{-23} +3 q^{-24} -14 q^{-25} -21 q^{-26} -22 q^{-27} +15 q^{-29} +17 q^{-30} +12 q^{-31} -16 q^{-33} -11 q^{-34} -4 q^{-35} +2 q^{-36} +9 q^{-37} +9 q^{-38} -2 q^{-39} -3 q^{-40} -3 q^{-41} -4 q^{-42} +4 q^{-44} + q^{-45} - q^{-48} - q^{-49} + q^{-50}$
6	$q^{138}-2 q^{137}+q^{136}+q^{133}-3 q^{132}+4 q^{131}-4 q^{130}+3 q^{129}-2 q^{128}+6 q^{126}-8 q^{125}+5 q^{124}-8 q^{123}+5 q^{122}-2 q^{121}+7 q^{120}+16 q^{119}-22 q^{118}-6 q^{117}-14 q^{116}+15 q^{115}+11 q^{114}+25 q^{113}+17 q^{112}-64 q^{111}-35 q^{110}-5 q^{109}+64 q^{108}+55 q^{107}+35 q^{106}-31 q^{105}-165 q^{104}-76 q^{103}+60 q^{102}+197 q^{101}+151 q^{100}+10 q^{99}-178 q^{98}-368 q^{97}-143 q^{96}+203 q^{95}+463 q^{94}+355 q^{93}-9 q^{92}-428 q^{91}-728 q^{90}-331 q^{89}+330 q^{88}+840 q^{87}+739 q^{86}+135 q^{85}-629 q^{84}-1192 q^{83}-730 q^{82}+232 q^{81}+1114 q^{80}+1205 q^{79}+526 q^{78}-537 q^{77}-1478 q^{76}-1176 q^{75}-139 q^{74}+1024 q^{73}+1417 q^{72}+922 q^{71}-169 q^{70}-1348 q^{69}-1309 q^{68}-475 q^{67}+672 q^{66}+1211 q^{65}+990 q^{64}+126 q^{63}-996 q^{62}-1084 q^{61}-516 q^{60}+420 q^{59}+881 q^{58}+801 q^{57}+171 q^{56}-769 q^{55}-841 q^{54}-417 q^{53}+355 q^{52}+718 q^{51}+681 q^{50}+170 q^{49}-677 q^{48}-773 q^{47}-444 q^{46}+266 q^{45}+649 q^{44}+716 q^{43}+307 q^{42}-516 q^{41}-740 q^{40}-589 q^{39}+32 q^{38}+483 q^{37}+740 q^{36}+520 q^{35}-220 q^{34}-578 q^{33}-681 q^{32}-258 q^{31}+184 q^{30}+612 q^{29}+653 q^{28}+119 q^{27}-268 q^{26}-601 q^{25}-455 q^{24}-162 q^{23}+315 q^{22}+594 q^{21}+363 q^{20}+100 q^{19}-328 q^{18}-438 q^{17}-406 q^{16}-59 q^{15}+318 q^{14}+373 q^{13}+355 q^{12}+33 q^{11}-180 q^{10}-389 q^9-311 q^8-45 q^7+126 q^6+326 q^5+247 q^4+151 q^3-111 q^2-260 q-231-167 q^{-1} +57 q^{-2} +150 q^{-3} +265 q^{-4} +165 q^{-5} +3 q^{-6} -104 q^{-7} -216 q^{-8} -153 q^{-9} -104 q^{-10} +87 q^{-11} +169 q^{-12} +150 q^{-13} +115 q^{-14} -25 q^{-15} -87 q^{-16} -184 q^{-17} -102 q^{-18} -16 q^{-19} +43 q^{-20} +129 q^{-21} +102 q^{-22} +83 q^{-23} -48 q^{-24} -68 q^{-25} -84 q^{-26} -89 q^{-27} -8 q^{-28} +29 q^{-29} +95 q^{-30} +44 q^{-31} +47 q^{-32} +4 q^{-33} -56 q^{-34} -52 q^{-35} -53 q^{-36} +4 q^{-37} -5 q^{-38} +48 q^{-39} +49 q^{-40} +19 q^{-41} +2 q^{-42} -26 q^{-43} -17 q^{-44} -45 q^{-45} -4 q^{-46} +12 q^{-47} +18 q^{-48} +20 q^{-49} +11 q^{-50} +11 q^{-51} -23 q^{-52} -11 q^{-53} -9 q^{-54} -3 q^{-55} +2 q^{-56} +7 q^{-57} +14 q^{-58} -3 q^{-59} -3 q^{-61} -3 q^{-62} -4 q^{-63} - q^{-64} +5 q^{-65} + q^{-67} - q^{-70} - q^{-71} + q^{-72}$

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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