10 46

10_45

10_47

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 10 46's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 46 at Knotilus!
	10_46 is also known as the pretzel knot P(5,3,2).

Knot presentations

Planar diagram presentation	X6271 X8493 X2837 X16,10,17,9 X14,5,15,6 X4,15,5,16 X18,12,19,11 X20,14,1,13 X10,18,11,17 X12,20,13,19
Gauss code	1, -3, 2, -6, 5, -1, 3, -2, 4, -9, 7, -10, 8, -5, 6, -4, 9, -7, 10, -8
Dowker-Thistlethwaite code	6 8 14 2 16 18 20 4 10 12
Conway Notation	[5,3,2]

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

[edit Notes on presentations of 10 46]

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 46"];

In[4]:=

PD[K]

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

Out[4]=

X6271 X8493 X2837 X16,10,17,9 X14,5,15,6 X4,15,5,16 X18,12,19,11 X20,14,1,13 X10,18,11,17 X12,20,13,19

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

6 8 14 2 16 18 20 4 10 12

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[5,3,2]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 3 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [3][-15] Hyperbolic Volume 7.717 A-Polynomial See Data:10 46/A-polynomial

[edit Notes for 10 46'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 3} Topological 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 3} Concordance 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 4} Rasmussen s-Invariant -6

[edit Notes for 10 46'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^3-4 t^2+5 t-5+5 t^{-1} -4 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^6-6 z^4+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	{ 31, 6 }
Jones polynomial	$q^{11}-2q^{10}+3q^{9}-4q^{8}+4q^{7}-5q^{6}+4q^{5}-3q^{4}+3q^{3}-q^{2}+q$
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^{-6} +z^6 a^{-4} -7 z^6 a^{-6} +z^6 a^{-8} +6 z^4 a^{-4} -17 z^4 a^{-6} +5 z^4 a^{-8} +11 z^2 a^{-4} -18 z^2 a^{-6} +7 z^2 a^{-8} +6 a^{-4} -8 a^{-6} +3 a^{-8} }
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^{-5} +z^9 a^{-7} +z^8 a^{-4} +4 z^8 a^{-6} +3 z^8 a^{-8} -5 z^7 a^{-5} -z^7 a^{-7} +4 z^7 a^{-9} -7 z^6 a^{-4} -23 z^6 a^{-6} -12 z^6 a^{-8} +4 z^6 a^{-10} +5 z^5 a^{-5} -12 z^5 a^{-7} -13 z^5 a^{-9} +4 z^5 a^{-11} +17 z^4 a^{-4} +42 z^4 a^{-6} +13 z^4 a^{-8} -9 z^4 a^{-10} +3 z^4 a^{-12} +5 z^3 a^{-5} +23 z^3 a^{-7} +9 z^3 a^{-9} -7 z^3 a^{-11} +2 z^3 a^{-13} -17 z^2 a^{-4} -29 z^2 a^{-6} -7 z^2 a^{-8} +2 z^2 a^{-10} -2 z^2 a^{-12} +z^2 a^{-14} -6 z a^{-5} -10 z a^{-7} -2 z a^{-9} +2 z a^{-11} +6 a^{-4} +8 a^{-6} +3 a^{-8} }
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^{-4} + q^{-6} +2 q^{-8} +2 q^{-10} + q^{-12} + q^{-14} -2 q^{-16} - q^{-18} -3 q^{-20} - q^{-22} + q^{-28} + q^{-32} }
The G2 invariant	$q^{-22}+3q^{-26}-2q^{-28}+3q^{-30}+6q^{-36}-6q^{-38}+9q^{-40}-3q^{-42}+q^{-44}+7q^{-46}-7q^{-48}+10q^{-50}-2q^{-52}+4q^{-56}-4q^{-58}+3q^{-60}+q^{-62}-4q^{-64}+2q^{-66}-2q^{-68}-q^{-70}-7q^{-74}+3q^{-76}-6q^{-78}+2q^{-80}-3q^{-82}-6q^{-84}+5q^{-86}-8q^{-88}+5q^{-90}-5q^{-92}-2q^{-94}+4q^{-96}-6q^{-98}+3q^{-100}-q^{-104}+3q^{-106}-2q^{-110}+4q^{-112}-q^{-114}+q^{-116}+q^{-118}-q^{-120}+3q^{-122}-q^{-124}+2q^{-126}+q^{-130}+q^{-134}-q^{-138}+3q^{-140}-3q^{-142}+3q^{-144}-q^{-146}-q^{-148}+q^{-150}-3q^{-152}+3q^{-154}-2q^{-156}+q^{-158}-2q^{-162}+2q^{-164}-2q^{-166}+2q^{-168}-q^{-170}-q^{-176}+q^{-178}-q^{-180}+q^{-182}$

Further Quantum Invariants

Further quantum knot invariants for 10_46.

A1 Invariants.

Weight	Invariant
1	$q^{-1}+2q^{-5}+q^{-9}-q^{-11}-q^{-13}-q^{-17}+q^{-19}-q^{-21}+q^{-23}$
2	$q^{2}-q^{-2}+2q^{-4}+2q^{-6}-2q^{-8}+q^{-10}+3q^{-12}-q^{-14}-q^{-16}+2q^{-18}-q^{-20}-2q^{-22}-2q^{-28}-q^{-30}+2q^{-32}-q^{-36}+2q^{-38}+q^{-40}-q^{-42}+q^{-46}-2q^{-54}+q^{-56}-q^{-60}+q^{-62}$
3	$q^{9}-q^{5}-q^{3}+2q+3q^{-1}-4q^{-5}-q^{-7}+4q^{-9}+5q^{-11}-2q^{-13}-4q^{-15}-q^{-17}+5q^{-19}+4q^{-21}-2q^{-23}-4q^{-25}+3q^{-29}+2q^{-31}-3q^{-33}-5q^{-35}-q^{-37}+2q^{-39}+2q^{-41}-3q^{-43}-q^{-45}+2q^{-47}+4q^{-49}-2q^{-51}-2q^{-53}+2q^{-55}+5q^{-57}-3q^{-59}-5q^{-61}+q^{-63}+6q^{-65}-5q^{-69}-3q^{-71}+2q^{-73}+5q^{-75}+2q^{-77}-4q^{-79}-7q^{-81}+3q^{-83}+6q^{-85}-q^{-87}-6q^{-89}+5q^{-93}+2q^{-95}-3q^{-97}-q^{-99}+q^{-101}+2q^{-103}-q^{-105}-q^{-107}-q^{-115}+q^{-117}$
4	$q^{20}-q^{16}-q^{14}-q^{12}+3q^{10}+3q^{8}+q^{6}-2q^{4}-7q^{2}-1+4q^{-2}+8q^{-4}+6q^{-6}-6q^{-8}-8q^{-10}-7q^{-12}+3q^{-14}+13q^{-16}+7q^{-18}+q^{-20}-11q^{-22}-11q^{-24}+2q^{-26}+9q^{-28}+14q^{-30}+3q^{-32}-10q^{-34}-11q^{-36}-8q^{-38}+7q^{-40}+13q^{-42}+5q^{-44}-5q^{-46}-15q^{-48}-9q^{-50}+5q^{-52}+13q^{-54}+12q^{-56}-5q^{-58}-15q^{-60}-9q^{-62}+5q^{-64}+18q^{-66}+9q^{-68}-9q^{-70}-16q^{-72}-7q^{-74}+13q^{-76}+13q^{-78}-3q^{-80}-11q^{-82}-8q^{-84}+8q^{-86}+10q^{-88}-5q^{-90}-10q^{-92}-2q^{-94}+13q^{-96}+13q^{-98}-8q^{-100}-18q^{-102}-8q^{-104}+12q^{-106}+21q^{-108}+5q^{-110}-15q^{-112}-19q^{-114}-7q^{-116}+14q^{-118}+20q^{-120}+9q^{-122}-11q^{-124}-25q^{-126}-9q^{-128}+15q^{-130}+24q^{-132}+8q^{-134}-21q^{-136}-21q^{-138}+q^{-140}+20q^{-142}+16q^{-144}-12q^{-146}-16q^{-148}-3q^{-150}+11q^{-152}+11q^{-154}-6q^{-156}-8q^{-158}-3q^{-160}+5q^{-162}+6q^{-164}-3q^{-166}-q^{-168}-q^{-170}+q^{-172}+2q^{-174}-2q^{-176}+q^{-178}-q^{-180}-q^{-186}+q^{-188}$
5	$q^{35}-q^{31}-q^{29}-q^{27}+3q^{23}+4q^{21}+q^{19}-2q^{17}-5q^{15}-7q^{13}-2q^{11}+6q^{9}+11q^{7}+9q^{5}+2q^{3}-10q-16q^{-1}-12q^{-3}+q^{-5}+15q^{-7}+21q^{-9}+14q^{-11}-2q^{-13}-19q^{-15}-26q^{-17}-14q^{-19}+7q^{-21}+23q^{-23}+30q^{-25}+16q^{-27}-9q^{-29}-28q^{-31}-28q^{-33}-13q^{-35}+11q^{-37}+32q^{-39}+31q^{-41}+11q^{-43}-15q^{-45}-33q^{-47}-33q^{-49}-11q^{-51}+19q^{-53}+36q^{-55}+32q^{-57}+8q^{-59}-24q^{-61}-43q^{-63}-35q^{-65}-q^{-67}+35q^{-69}+51q^{-71}+29q^{-73}-13q^{-75}-51q^{-77}-52q^{-79}-14q^{-81}+38q^{-83}+62q^{-85}+38q^{-87}-16q^{-89}-61q^{-91}-59q^{-93}-5q^{-95}+49q^{-97}+65q^{-99}+28q^{-101}-32q^{-103}-66q^{-105}-42q^{-107}+14q^{-109}+55q^{-111}+51q^{-113}+5q^{-115}-42q^{-117}-50q^{-119}-14q^{-121}+27q^{-123}+42q^{-125}+20q^{-127}-15q^{-129}-28q^{-131}-12q^{-133}+11q^{-135}+19q^{-137}-2q^{-139}-21q^{-141}-16q^{-143}+12q^{-145}+35q^{-147}+28q^{-149}-14q^{-151}-51q^{-153}-46q^{-155}+47q^{-159}+62q^{-161}+27q^{-163}-30q^{-165}-63q^{-167}-53q^{-169}-5q^{-171}+44q^{-173}+66q^{-175}+46q^{-177}-10q^{-179}-61q^{-181}-69q^{-183}-28q^{-185}+38q^{-187}+79q^{-189}+55q^{-191}-14q^{-193}-69q^{-195}-66q^{-197}-5q^{-199}+53q^{-201}+62q^{-203}+16q^{-205}-39q^{-207}-52q^{-209}-15q^{-211}+30q^{-213}+38q^{-215}+10q^{-217}-23q^{-219}-32q^{-221}-3q^{-223}+23q^{-225}+21q^{-227}-q^{-229}-18q^{-231}-15q^{-233}+q^{-235}+15q^{-237}+10q^{-239}-3q^{-241}-10q^{-243}-5q^{-245}+2q^{-247}+5q^{-249}+3q^{-251}-q^{-253}-3q^{-255}+q^{-259}+q^{-261}-q^{-267}-q^{-273}+q^{-275}$
6	$q^{54}-q^{50}-q^{48}-q^{46}+4q^{40}+4q^{38}+q^{36}-2q^{34}-5q^{32}-6q^{30}-8q^{28}+q^{26}+8q^{24}+13q^{22}+12q^{20}+6q^{18}-4q^{16}-21q^{14}-21q^{12}-16q^{10}+17q^{6}+31q^{4}+33q^{2}+12-9q^{-2}-34q^{-4}-42q^{-6}-36q^{-8}-6q^{-10}+29q^{-12}+48q^{-14}+55q^{-16}+30q^{-18}-6q^{-20}-48q^{-22}-64q^{-24}-52q^{-26}-21q^{-28}+29q^{-30}+64q^{-32}+75q^{-34}+47q^{-36}+3q^{-38}-44q^{-40}-78q^{-42}-74q^{-44}-39q^{-46}+18q^{-48}+64q^{-50}+87q^{-52}+76q^{-54}+24q^{-56}-37q^{-58}-87q^{-60}-99q^{-62}-70q^{-64}-8q^{-66}+69q^{-68}+116q^{-70}+112q^{-72}+51q^{-74}-37q^{-76}-118q^{-78}-151q^{-80}-102q^{-82}+2q^{-84}+114q^{-86}+169q^{-88}+147q^{-90}+40q^{-92}-103q^{-94}-191q^{-96}-181q^{-98}-68q^{-100}+80q^{-102}+201q^{-104}+210q^{-106}+94q^{-108}-73q^{-110}-203q^{-112}-217q^{-114}-116q^{-116}+65q^{-118}+205q^{-120}+220q^{-122}+113q^{-124}-60q^{-126}-195q^{-128}-222q^{-130}-105q^{-132}+69q^{-134}+191q^{-136}+204q^{-138}+97q^{-140}-70q^{-142}-197q^{-144}-187q^{-146}-66q^{-148}+86q^{-150}+188q^{-152}+175q^{-154}+49q^{-156}-111q^{-158}-187q^{-160}-147q^{-162}-20q^{-164}+118q^{-166}+181q^{-168}+122q^{-170}-17q^{-172}-128q^{-174}-154q^{-176}-84q^{-178}+30q^{-180}+117q^{-182}+114q^{-184}+32q^{-186}-45q^{-188}-75q^{-190}-44q^{-192}+8q^{-194}+39q^{-196}+14q^{-198}-39q^{-200}-53q^{-202}-10q^{-204}+69q^{-206}+113q^{-208}+87q^{-210}-21q^{-212}-137q^{-214}-176q^{-216}-106q^{-218}+45q^{-220}+174q^{-222}+214q^{-224}+127q^{-226}-31q^{-228}-178q^{-230}-230q^{-232}-158q^{-234}-11q^{-236}+151q^{-238}+236q^{-240}+212q^{-242}+63q^{-244}-122q^{-246}-249q^{-248}-251q^{-250}-108q^{-252}+98q^{-254}+267q^{-256}+268q^{-258}+119q^{-260}-100q^{-262}-261q^{-264}-262q^{-266}-108q^{-268}+117q^{-270}+242q^{-272}+218q^{-274}+66q^{-276}-116q^{-278}-209q^{-280}-162q^{-282}-10q^{-284}+114q^{-286}+153q^{-288}+90q^{-290}-19q^{-292}-97q^{-294}-94q^{-296}-22q^{-298}+32q^{-300}+54q^{-302}+34q^{-304}-3q^{-306}-25q^{-308}-19q^{-310}+10q^{-312}+8q^{-314}-q^{-316}-9q^{-318}-11q^{-320}-q^{-322}+10q^{-324}+21q^{-326}+3q^{-328}-12q^{-330}-14q^{-332}-7q^{-334}+4q^{-336}+9q^{-338}+11q^{-340}-2q^{-342}-7q^{-344}-5q^{-346}-2q^{-348}+4q^{-350}+2q^{-352}+3q^{-354}-2q^{-356}-2q^{-358}+3q^{-364}-q^{-366}-q^{-370}-q^{-376}+q^{-378}$

A2 Invariants.

Weight	Invariant
1,0	$q^{-4}+q^{-6}+2q^{-8}+2q^{-10}+q^{-12}+q^{-14}-2q^{-16}-q^{-18}-3q^{-20}-q^{-22}+q^{-28}+q^{-32}$
1,1	$q^{-4}+6q^{-8}-4q^{-10}+14q^{-12}-12q^{-14}+24q^{-16}-18q^{-18}+20q^{-20}-16q^{-22}+6q^{-24}-2q^{-26}-14q^{-28}+10q^{-30}-26q^{-32}+22q^{-34}-29q^{-36}+26q^{-38}-22q^{-40}+22q^{-42}-9q^{-44}+10q^{-46}+5q^{-52}-2q^{-54}-2q^{-62}-6q^{-66}+5q^{-68}-6q^{-70}+8q^{-72}-6q^{-74}+5q^{-76}-4q^{-78}+4q^{-80}-4q^{-82}+3q^{-84}-2q^{-86}+2q^{-88}-2q^{-90}+q^{-92}$
2,0	$q^{-4}+q^{-6}+q^{-8}+q^{-10}+3q^{-12}+3q^{-14}+2q^{-16}+q^{-18}+3q^{-20}+q^{-22}-q^{-24}-2q^{-26}-q^{-28}-3q^{-30}-4q^{-32}-4q^{-34}-4q^{-36}-4q^{-38}-2q^{-40}+q^{-42}+2q^{-44}+4q^{-46}+6q^{-48}+5q^{-50}+q^{-52}+q^{-54}-2q^{-60}-q^{-62}-q^{-64}-q^{-66}-2q^{-68}-q^{-70}+q^{-76}+q^{-80}$

A3 Invariants.

Weight	Invariant
0,1,0	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^{-8} +3 q^{-12} +3 q^{-14} +4 q^{-16} +6 q^{-18} +5 q^{-20} +2 q^{-22} + q^{-24} -4 q^{-26} -8 q^{-28} -8 q^{-30} -8 q^{-32} -5 q^{-34} + q^{-38} +3 q^{-40} +5 q^{-42} +3 q^{-44} +2 q^{-46} + q^{-48} +2 q^{-50} - q^{-54} -2 q^{-60} + q^{-64} -2 q^{-66} +2 q^{-70} - q^{-72} - q^{-74} + q^{-76} }
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^{-7} + q^{-9} +3 q^{-11} +2 q^{-13} +4 q^{-15} + q^{-17} + q^{-19} -2 q^{-21} -3 q^{-23} -4 q^{-25} -3 q^{-27} - q^{-29} - q^{-31} +2 q^{-33} +2 q^{-37} + q^{-41} }
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^{-10} +6 q^{-14} +2 q^{-16} +13 q^{-18} +8 q^{-20} +14 q^{-22} +14 q^{-24} +6 q^{-26} +12 q^{-28} -10 q^{-30} -27 q^{-34} -13 q^{-36} -30 q^{-38} -17 q^{-40} -13 q^{-42} -14 q^{-44} +14 q^{-46} -8 q^{-48} +32 q^{-50} + q^{-52} +27 q^{-54} +5 q^{-56} +7 q^{-58} +9 q^{-60} -10 q^{-62} +7 q^{-64} -17 q^{-66} +8 q^{-68} -13 q^{-70} +4 q^{-72} -2 q^{-74} -3 q^{-76} +5 q^{-78} -4 q^{-80} +5 q^{-82} -4 q^{-84} +2 q^{-86} -3 q^{-88} - q^{-90} - q^{-92} - q^{-94} +4 q^{-96} - q^{-98} + q^{-100} - q^{-104} +2 q^{-106} - q^{-108} - q^{-110} +2 q^{-112} -2 q^{-114} + q^{-116} + q^{-118} -2 q^{-120} + q^{-122} }

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^{-14} + q^{-16} +3 q^{-18} +5 q^{-20} +7 q^{-22} +9 q^{-24} +11 q^{-26} +10 q^{-28} +8 q^{-30} +3 q^{-32} -3 q^{-34} -10 q^{-36} -16 q^{-38} -21 q^{-40} -19 q^{-42} -16 q^{-44} -9 q^{-46} -3 q^{-48} +5 q^{-50} +12 q^{-52} +12 q^{-54} +11 q^{-56} +11 q^{-58} +9 q^{-60} +2 q^{-62} +2 q^{-64} -3 q^{-68} -4 q^{-70} -2 q^{-72} -3 q^{-74} -4 q^{-76} - q^{-78} + q^{-80} - q^{-82} - q^{-84} +2 q^{-86} + q^{-88} - q^{-90} + q^{-94} }
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^{-10} + q^{-12} +3 q^{-14} +3 q^{-16} +4 q^{-18} +4 q^{-20} +2 q^{-22} + q^{-24} -3 q^{-26} -3 q^{-28} -6 q^{-30} -4 q^{-32} -4 q^{-34} - q^{-36} + q^{-40} +2 q^{-42} + q^{-44} +2 q^{-46} + q^{-50} }

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^{-8} +3 q^{-12} - q^{-14} +4 q^{-16} -2 q^{-18} +5 q^{-20} -2 q^{-22} +3 q^{-24} -2 q^{-26} -4 q^{-32} +3 q^{-34} -6 q^{-36} +5 q^{-38} -7 q^{-40} +5 q^{-42} -5 q^{-44} +4 q^{-46} -3 q^{-48} +2 q^{-50} - q^{-54} +2 q^{-56} -2 q^{-58} +2 q^{-60} -2 q^{-62} +3 q^{-64} -2 q^{-66} +2 q^{-68} -2 q^{-70} + q^{-72} - q^{-74} + q^{-76} }
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^{-10} +3 q^{-18} +2 q^{-20} +4 q^{-26} +4 q^{-28} +2 q^{-30} -2 q^{-32} + q^{-34} +3 q^{-36} +2 q^{-38} -3 q^{-40} -5 q^{-42} -2 q^{-44} -2 q^{-48} -6 q^{-50} -5 q^{-52} - q^{-54} + q^{-56} - q^{-58} - q^{-60} +3 q^{-64} + q^{-66} + q^{-70} +4 q^{-72} +2 q^{-74} - q^{-76} - q^{-78} +2 q^{-80} +3 q^{-82} -2 q^{-86} - q^{-88} + q^{-90} + q^{-92} - q^{-94} -2 q^{-96} - q^{-98} + q^{-100} +2 q^{-102} - q^{-104} -2 q^{-106} - q^{-108} + q^{-110} +2 q^{-112} - q^{-116} - q^{-118} + q^{-122} }

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^{-14} +3 q^{-18} + q^{-20} +7 q^{-22} +3 q^{-24} +9 q^{-26} +4 q^{-28} +9 q^{-30} + q^{-32} +2 q^{-34} -5 q^{-36} -6 q^{-38} -10 q^{-40} -12 q^{-42} -8 q^{-44} -10 q^{-46} - q^{-48} -6 q^{-50} +6 q^{-52} +10 q^{-56} +8 q^{-60} - q^{-62} +5 q^{-64} - q^{-66} +2 q^{-68} - q^{-70} + q^{-74} - q^{-76} + q^{-78} -2 q^{-80} + q^{-82} -2 q^{-84} + q^{-86} -2 q^{-88} +2 q^{-90} -2 q^{-92} + q^{-94} - q^{-96} +2 q^{-98} - q^{-100} - q^{-104} + q^{-106} }

G2 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^{-22} +3 q^{-26} -2 q^{-28} +3 q^{-30} +6 q^{-36} -6 q^{-38} +9 q^{-40} -3 q^{-42} + q^{-44} +7 q^{-46} -7 q^{-48} +10 q^{-50} -2 q^{-52} +4 q^{-56} -4 q^{-58} +3 q^{-60} + q^{-62} -4 q^{-64} +2 q^{-66} -2 q^{-68} - q^{-70} -7 q^{-74} +3 q^{-76} -6 q^{-78} +2 q^{-80} -3 q^{-82} -6 q^{-84} +5 q^{-86} -8 q^{-88} +5 q^{-90} -5 q^{-92} -2 q^{-94} +4 q^{-96} -6 q^{-98} +3 q^{-100} - q^{-104} +3 q^{-106} -2 q^{-110} +4 q^{-112} - q^{-114} + q^{-116} + q^{-118} - q^{-120} +3 q^{-122} - q^{-124} +2 q^{-126} + q^{-130} + q^{-134} - q^{-138} +3 q^{-140} -3 q^{-142} +3 q^{-144} - q^{-146} - q^{-148} + q^{-150} -3 q^{-152} +3 q^{-154} -2 q^{-156} + q^{-158} -2 q^{-162} +2 q^{-164} -2 q^{-166} +2 q^{-168} - q^{-170} - q^{-176} + q^{-178} - q^{-180} + q^{-182} }

.

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 46"];

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^3-4 t^2+5 t-5+5 t^{-1} -4 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^6-6 z^4+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]=

{ 31, 6 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{11}-2q^{10}+3q^{9}-4q^{8}+4q^{7}-5q^{6}+4q^{5}-3q^{4}+3q^{3}-q^{2}+q$

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^{-6} +z^6 a^{-4} -7 z^6 a^{-6} +z^6 a^{-8} +6 z^4 a^{-4} -17 z^4 a^{-6} +5 z^4 a^{-8} +11 z^2 a^{-4} -18 z^2 a^{-6} +7 z^2 a^{-8} +6 a^{-4} -8 a^{-6} +3 a^{-8} }

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^{-5} +z^9 a^{-7} +z^8 a^{-4} +4 z^8 a^{-6} +3 z^8 a^{-8} -5 z^7 a^{-5} -z^7 a^{-7} +4 z^7 a^{-9} -7 z^6 a^{-4} -23 z^6 a^{-6} -12 z^6 a^{-8} +4 z^6 a^{-10} +5 z^5 a^{-5} -12 z^5 a^{-7} -13 z^5 a^{-9} +4 z^5 a^{-11} +17 z^4 a^{-4} +42 z^4 a^{-6} +13 z^4 a^{-8} -9 z^4 a^{-10} +3 z^4 a^{-12} +5 z^3 a^{-5} +23 z^3 a^{-7} +9 z^3 a^{-9} -7 z^3 a^{-11} +2 z^3 a^{-13} -17 z^2 a^{-4} -29 z^2 a^{-6} -7 z^2 a^{-8} +2 z^2 a^{-10} -2 z^2 a^{-12} +z^2 a^{-14} -6 z a^{-5} -10 z a^{-7} -2 z a^{-9} +2 z a^{-11} +6 a^{-4} +8 a^{-6} +3 a^{-8} }

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n60,}

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 46"];

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^3-4 t^2+5 t-5+5 t^{-1} -4 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^{11}-2 q^{10}+3 q^9-4 q^8+4 q^7-5 q^6+4 q^5-3 q^4+3 q^3-q^2+q} }

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]=

{K11n60,}

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:

(0, -4)

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 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 -32} 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 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 -96} 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 48} 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 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 \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{544}{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 256} 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 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 512} 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 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 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 2224} 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 -176} $2192$ 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 304} 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 112}

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=} 6 is the signature of 10 46. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -2 -1 0 1 2 3 4 5 6 7 8 χ 23                     1 1 21                   1   -1 19                 2 1   1 17               2 1     -1 15             2 2       0 13           3 2         -1 11         1 2           -1 9       2 3             1 7     1 1               0 5   1 3                 2 3                       0 1 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}} $i=5$ 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 i=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 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}} 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} 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}} 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^{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} 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=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}^{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=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^{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=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}\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=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 {\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^{30}-2 q^{29}+q^{28}+2 q^{27}-5 q^{26}+3 q^{25}+2 q^{24}-5 q^{23}+4 q^{22}+q^{21}-6 q^{20}+6 q^{19}+2 q^{18}-9 q^{17}+7 q^{16}+4 q^{15}-12 q^{14}+6 q^{13}+6 q^{12}-12 q^{11}+4 q^{10}+7 q^9-9 q^8+q^7+7 q^6-5 q^5-q^4+4 q^3-q^2-q+1}
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^{57}-2 q^{56}+q^{55}+q^{53}-3 q^{52}+q^{51}+3 q^{50}-5 q^{48}-q^{47}+8 q^{46}+3 q^{45}-10 q^{44}-7 q^{43}+13 q^{42}+10 q^{41}-13 q^{40}-17 q^{39}+16 q^{38}+16 q^{37}-10 q^{36}-20 q^{35}+11 q^{34}+14 q^{33}-5 q^{32}-14 q^{31}+6 q^{30}+8 q^{29}-3 q^{28}-6 q^{27}+3 q^{26}+4 q^{25}-3 q^{24}+q^{22}+q^{21}-5 q^{20}+5 q^{19}+q^{18}-2 q^{17}-9 q^{16}+7 q^{15}+6 q^{14}-q^{13}-12 q^{12}+3 q^{11}+8 q^{10}+5 q^9-11 q^8-3 q^7+5 q^6+7 q^5-4 q^4-4 q^3+4 q- q^{-1} - q^{-2} + q^{-3} }
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^{92}-2 q^{91}+q^{90}-q^{88}+3 q^{87}-5 q^{86}+5 q^{85}-q^{84}-3 q^{83}+3 q^{82}-7 q^{81}+14 q^{80}-2 q^{79}-11 q^{78}-2 q^{77}-5 q^{76}+31 q^{75}-2 q^{74}-25 q^{73}-15 q^{72}-q^{71}+59 q^{70}+2 q^{69}-44 q^{68}-37 q^{67}-q^{66}+88 q^{65}+18 q^{64}-53 q^{63}-61 q^{62}-17 q^{61}+102 q^{60}+38 q^{59}-42 q^{58}-67 q^{57}-38 q^{56}+90 q^{55}+42 q^{54}-22 q^{53}-51 q^{52}-47 q^{51}+70 q^{50}+32 q^{49}-12 q^{48}-30 q^{47}-47 q^{46}+55 q^{45}+24 q^{44}-7 q^{43}-15 q^{42}-49 q^{41}+39 q^{40}+21 q^{39}+q^{38}+q^{37}-49 q^{36}+19 q^{35}+12 q^{34}+8 q^{33}+19 q^{32}-40 q^{31}+6 q^{30}-2 q^{29}+2 q^{28}+29 q^{27}-23 q^{26}+7 q^{25}-10 q^{24}-12 q^{23}+23 q^{22}-13 q^{21}+17 q^{20}-2 q^{19}-18 q^{18}+8 q^{17}-16 q^{16}+18 q^{15}+11 q^{14}-7 q^{13}+3 q^{12}-23 q^{11}+5 q^{10}+11 q^9+5 q^8+9 q^7-17 q^6-5 q^5+q^4+4 q^3+11 q^2-5 q-3-3 q^{-1} - q^{-2} +5 q^{-3} - q^{-6} - q^{-7} + q^{-8} }
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^{135}-2 q^{134}+q^{133}-q^{131}+q^{130}+q^{129}-q^{128}+q^{127}-q^{126}-4 q^{125}+3 q^{124}+5 q^{123}+q^{122}-2 q^{121}-8 q^{120}-9 q^{119}+10 q^{118}+18 q^{117}+6 q^{116}-16 q^{115}-24 q^{114}-12 q^{113}+27 q^{112}+40 q^{111}+8 q^{110}-42 q^{109}-53 q^{108}-3 q^{107}+60 q^{106}+68 q^{105}-87 q^{103}-90 q^{102}+10 q^{101}+115 q^{100}+114 q^{99}-9 q^{98}-145 q^{97}-151 q^{96}+7 q^{95}+170 q^{94}+183 q^{93}+15 q^{92}-186 q^{91}-217 q^{90}-34 q^{89}+178 q^{88}+234 q^{87}+71 q^{86}-166 q^{85}-239 q^{84}-83 q^{83}+130 q^{82}+224 q^{81}+104 q^{80}-109 q^{79}-204 q^{78}-98 q^{77}+83 q^{76}+178 q^{75}+99 q^{74}-72 q^{73}-162 q^{72}-91 q^{71}+60 q^{70}+150 q^{69}+94 q^{68}-53 q^{67}-141 q^{66}-99 q^{65}+37 q^{64}+134 q^{63}+107 q^{62}-18 q^{61}-119 q^{60}-114 q^{59}-4 q^{58}+98 q^{57}+115 q^{56}+29 q^{55}-73 q^{54}-110 q^{53}-45 q^{52}+42 q^{51}+91 q^{50}+63 q^{49}-13 q^{48}-73 q^{47}-61 q^{46}-12 q^{45}+37 q^{44}+61 q^{43}+32 q^{42}-19 q^{41}-37 q^{40}-36 q^{39}-15 q^{38}+23 q^{37}+33 q^{36}+19 q^{35}+5 q^{34}-14 q^{33}-31 q^{32}-13 q^{31}-q^{30}+11 q^{29}+24 q^{28}+18 q^{27}-7 q^{26}-9 q^{25}-18 q^{24}-19 q^{23}+2 q^{22}+18 q^{21}+11 q^{20}+17 q^{19}+2 q^{18}-18 q^{17}-19 q^{16}-6 q^{15}-4 q^{14}+17 q^{13}+21 q^{12}+7 q^{11}-5 q^{10}-13 q^9-20 q^8-4 q^7+9 q^6+14 q^5+12 q^4+3 q^3-13 q^2-10 q-5+ q^{-1} +8 q^{-2} +9 q^{-3} - q^{-4} -3 q^{-5} -3 q^{-6} -4 q^{-7} +4 q^{-9} + q^{-10} - q^{-13} - q^{-14} + q^{-15} }
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^{186}-2 q^{185}+q^{184}-q^{182}+q^{181}-q^{180}+5 q^{179}-5 q^{178}+q^{177}-2 q^{176}-q^{175}+6 q^{174}-2 q^{173}+7 q^{172}-11 q^{171}-2 q^{170}-4 q^{169}+4 q^{168}+19 q^{167}-4 q^{166}+2 q^{165}-22 q^{164}-9 q^{163}-2 q^{162}+19 q^{161}+37 q^{160}-15 q^{159}-9 q^{158}-32 q^{157}-7 q^{156}+6 q^{155}+28 q^{154}+39 q^{153}-44 q^{152}-15 q^{151}-10 q^{150}+30 q^{149}+26 q^{148}+6 q^{147}-15 q^{146}-116 q^{145}-18 q^{144}+68 q^{143}+139 q^{142}+89 q^{141}-33 q^{140}-139 q^{139}-268 q^{138}-65 q^{137}+161 q^{136}+321 q^{135}+241 q^{134}-9 q^{133}-264 q^{132}-493 q^{131}-219 q^{130}+162 q^{129}+482 q^{128}+460 q^{127}+140 q^{126}-265 q^{125}-662 q^{124}-425 q^{123}+19 q^{122}+484 q^{121}+587 q^{120}+325 q^{119}-116 q^{118}-638 q^{117}-510 q^{116}-143 q^{115}+337 q^{114}+515 q^{113}+377 q^{112}+31 q^{111}-480 q^{110}-423 q^{109}-183 q^{108}+208 q^{107}+364 q^{106}+307 q^{105}+70 q^{104}-364 q^{103}-315 q^{102}-157 q^{101}+164 q^{100}+285 q^{99}+264 q^{98}+84 q^{97}-311 q^{96}-290 q^{95}-188 q^{94}+112 q^{93}+254 q^{92}+294 q^{91}+161 q^{90}-229 q^{89}-287 q^{88}-275 q^{87}-2 q^{86}+184 q^{85}+320 q^{84}+272 q^{83}-90 q^{82}-228 q^{81}-338 q^{80}-140 q^{79}+57 q^{78}+280 q^{77}+348 q^{76}+70 q^{75}-102 q^{74}-325 q^{73}-242 q^{72}-95 q^{71}+159 q^{70}+338 q^{69}+197 q^{68}+65 q^{67}-218 q^{66}-255 q^{65}-217 q^{64}-15 q^{63}+221 q^{62}+224 q^{61}+200 q^{60}-45 q^{59}-148 q^{58}-232 q^{57}-155 q^{56}+40 q^{55}+123 q^{54}+214 q^{53}+85 q^{52}+19 q^{51}-116 q^{50}-164 q^{49}-81 q^{48}-25 q^{47}+101 q^{46}+75 q^{45}+107 q^{44}+27 q^{43}-57 q^{42}-59 q^{41}-80 q^{40}-11 q^{39}-29 q^{38}+58 q^{37}+60 q^{36}+24 q^{35}+29 q^{34}-19 q^{33}-7 q^{32}-76 q^{31}-19 q^{30}-2 q^{29}-5 q^{28}+41 q^{27}+31 q^{26}+54 q^{25}-24 q^{24}-8 q^{23}-25 q^{22}-51 q^{21}-16 q^{20}-4 q^{19}+50 q^{18}+10 q^{17}+39 q^{16}+19 q^{15}-23 q^{14}-27 q^{13}-39 q^{12}-21 q^{10}+27 q^9+35 q^8+19 q^7+9 q^6-14 q^5-7 q^4-40 q^3-8 q^2+5 q+13+17 q^{-1} +11 q^{-2} +14 q^{-3} -19 q^{-4} -10 q^{-5} -9 q^{-6} -4 q^{-7} + q^{-8} +6 q^{-9} +14 q^{-10} -2 q^{-11} -3 q^{-13} -3 q^{-14} -4 q^{-15} - q^{-16} +5 q^{-17} + q^{-19} - q^{-22} - q^{-23} + q^{-24} }
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^{245}-2 q^{244}+q^{243}-q^{241}+q^{240}-q^{239}+3 q^{238}+q^{237}-5 q^{236}+q^{234}+2 q^{233}+2 q^{232}-5 q^{231}+4 q^{230}+q^{229}-9 q^{228}-q^{227}+7 q^{226}+8 q^{225}+3 q^{224}-11 q^{223}-2 q^{222}-4 q^{221}-10 q^{220}+5 q^{219}+13 q^{218}+17 q^{217}+9 q^{216}-13 q^{215}-17 q^{214}-21 q^{213}-18 q^{212}+6 q^{211}+19 q^{210}+37 q^{209}+43 q^{208}+14 q^{207}-21 q^{206}-66 q^{205}-85 q^{204}-51 q^{203}+8 q^{202}+100 q^{201}+161 q^{200}+135 q^{199}+17 q^{198}-155 q^{197}-282 q^{196}-250 q^{195}-62 q^{194}+214 q^{193}+433 q^{192}+424 q^{191}+156 q^{190}-272 q^{189}-637 q^{188}-658 q^{187}-289 q^{186}+315 q^{185}+845 q^{184}+942 q^{183}+498 q^{182}-294 q^{181}-1049 q^{180}-1273 q^{179}-787 q^{178}+195 q^{177}+1210 q^{176}+1602 q^{175}+1130 q^{174}+4 q^{173}-1247 q^{172}-1881 q^{171}-1523 q^{170}-310 q^{169}+1183 q^{168}+2065 q^{167}+1849 q^{166}+655 q^{165}-965 q^{164}-2080 q^{163}-2090 q^{162}-998 q^{161}+686 q^{160}+1958 q^{159}+2159 q^{158}+1222 q^{157}-386 q^{156}-1716 q^{155}-2067 q^{154}-1312 q^{153}+162 q^{152}+1450 q^{151}+1856 q^{150}+1248 q^{149}-51 q^{148}-1220 q^{147}-1617 q^{146}-1094 q^{145}+60 q^{144}+1084 q^{143}+1401 q^{142}+913 q^{141}-112 q^{140}-1023 q^{139}-1285 q^{138}-793 q^{137}+190 q^{136}+1018 q^{135}+1213 q^{134}+730 q^{133}-182 q^{132}-983 q^{131}-1207 q^{130}-761 q^{129}+132 q^{128}+926 q^{127}+1171 q^{126}+800 q^{125}-3 q^{124}-779 q^{123}-1116 q^{122}-863 q^{121}-152 q^{120}+610 q^{119}+1011 q^{118}+881 q^{117}+307 q^{116}-388 q^{115}-856 q^{114}-881 q^{113}-461 q^{112}+169 q^{111}+683 q^{110}+832 q^{109}+569 q^{108}+61 q^{107}-468 q^{106}-741 q^{105}-664 q^{104}-278 q^{103}+243 q^{102}+617 q^{101}+690 q^{100}+458 q^{99}+9 q^{98}-421 q^{97}-669 q^{96}-620 q^{95}-244 q^{94}+210 q^{93}+559 q^{92}+675 q^{91}+464 q^{90}+65 q^{89}-386 q^{88}-678 q^{87}-608 q^{86}-299 q^{85}+141 q^{84}+547 q^{83}+671 q^{82}+517 q^{81}+120 q^{80}-358 q^{79}-605 q^{78}-627 q^{77}-368 q^{76}+92 q^{75}+455 q^{74}+635 q^{73}+526 q^{72}+152 q^{71}-204 q^{70}-510 q^{69}-593 q^{68}-358 q^{67}-36 q^{66}+315 q^{65}+510 q^{64}+432 q^{63}+259 q^{62}-61 q^{61}-353 q^{60}-416 q^{59}-357 q^{58}-129 q^{57}+136 q^{56}+262 q^{55}+356 q^{54}+261 q^{53}+43 q^{52}-96 q^{51}-247 q^{50}-252 q^{49}-150 q^{48}-69 q^{47}+97 q^{46}+176 q^{45}+147 q^{44}+150 q^{43}+33 q^{42}-55 q^{41}-78 q^{40}-141 q^{39}-85 q^{38}-43 q^{37}-31 q^{36}+76 q^{35}+78 q^{34}+71 q^{33}+83 q^{32}+10 q^{31}-q^{30}-36 q^{29}-109 q^{28}-55 q^{27}-48 q^{26}-21 q^{25}+48 q^{24}+44 q^{23}+83 q^{22}+81 q^{21}-q^{20}-7 q^{19}-51 q^{18}-80 q^{17}-44 q^{16}-50 q^{15}+5 q^{14}+59 q^{13}+42 q^{12}+67 q^{11}+36 q^{10}-8 q^9-13 q^8-55 q^7-55 q^6-23 q^5-22 q^4+27 q^3+40 q^2+28 q+40+9 q^{-1} -13 q^{-2} -20 q^{-3} -42 q^{-4} -19 q^{-5} -3 q^{-6} -2 q^{-7} +21 q^{-8} +21 q^{-9} +17 q^{-10} +13 q^{-11} -13 q^{-12} -11 q^{-13} -8 q^{-14} -13 q^{-15} -4 q^{-16} + q^{-17} +7 q^{-18} +13 q^{-19} + q^{-20} + q^{-22} -4 q^{-23} -3 q^{-24} -4 q^{-25} - q^{-26} +4 q^{-27} + q^{-28} + q^{-30} - q^{-33} - q^{-34} + q^{-35} }

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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