10 157

10_156

10_158

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10 157 Further Notes and Views

Knot presentations

Planar diagram presentation	X1627 X10,4,11,3 X16,11,17,12 X7,15,8,14 X15,9,16,8 X13,1,14,20 X19,13,20,12 X18,6,19,5 X2,10,3,9 X4,18,5,17
Gauss code	-1, -9, 2, -10, 8, 1, -4, 5, 9, -2, 3, 7, -6, 4, -5, -3, 10, -8, -7, 6
Dowker-Thistlethwaite code	6 -10 -18 14 -2 -16 20 8 -4 12
Conway Notation	[-3:20:20]

Minimum Braid Representative: Length is 10, width is 3. Braid index is 3.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 2 Maximal Thurston-Bennequin number [3][-13] Hyperbolic Volume 12.6653 A-Polynomial See Data:10 157/A-polynomial

[edit Notes for 10 157'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 10 157's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+6t^{2}-11t+13-11t^{-1}+6t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}+4z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{7,t+1\}$
Determinant and Signature	{ 49, 4 }
Jones polynomial	$q^{10}-4q^{9}+6q^{8}-8q^{7}+9q^{6}-8q^{5}+7q^{4}-4q^{3}+2q^{2}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-6}+2z^{4}a^{-4}-3z^{4}a^{-6}+z^{4}a^{-8}+5z^{2}a^{-4}-2z^{2}a^{-6}+z^{2}a^{-8}+2a^{-4}-a^{-8}$
Kauffman polynomial (db, data sources)	$z^{8}a^{-6}+z^{8}a^{-8}+z^{7}a^{-5}+5z^{7}a^{-7}+4z^{7}a^{-9}+2z^{6}a^{-6}+8z^{6}a^{-8}+6z^{6}a^{-10}+z^{5}a^{-5}-3z^{5}a^{-7}+4z^{5}a^{-11}+3z^{4}a^{-4}-3z^{4}a^{-6}-15z^{4}a^{-8}-8z^{4}a^{-10}+z^{4}a^{-12}-2z^{3}a^{-5}-6z^{3}a^{-7}-8z^{3}a^{-9}-4z^{3}a^{-11}-5z^{2}a^{-4}+7z^{2}a^{-8}+2z^{2}a^{-10}+4za^{-7}+4za^{-9}+2a^{-4}-a^{-8}$
The A2 invariant	$2q^{-6}-q^{-8}+2q^{-10}+3q^{-16}-q^{-18}+2q^{-20}-2q^{-22}-q^{-24}-2q^{-28}+q^{-30}$
The G2 invariant	$q^{-28}-2q^{-32}+12q^{-34}-18q^{-36}+19q^{-38}-9q^{-40}-10q^{-42}+42q^{-44}-60q^{-46}+66q^{-48}-44q^{-50}-4q^{-52}+56q^{-54}-96q^{-56}+101q^{-58}-67q^{-60}+7q^{-62}+49q^{-64}-79q^{-66}+74q^{-68}-31q^{-70}-19q^{-72}+61q^{-74}-66q^{-76}+37q^{-78}+18q^{-80}-70q^{-82}+105q^{-84}-99q^{-86}+61q^{-88}+4q^{-90}-72q^{-92}+119q^{-94}-134q^{-96}+99q^{-98}-38q^{-100}-37q^{-102}+85q^{-104}-102q^{-106}+73q^{-108}-17q^{-110}-38q^{-112}+64q^{-114}-58q^{-116}+13q^{-118}+43q^{-120}-81q^{-122}+85q^{-124}-51q^{-126}+52q^{-130}-83q^{-132}+85q^{-134}-59q^{-136}+20q^{-138}+14q^{-140}-40q^{-142}+45q^{-144}-34q^{-146}+22q^{-148}-5q^{-150}-4q^{-152}+8q^{-154}-10q^{-156}+6q^{-158}-3q^{-160}+q^{-162}$

Further Quantum Invariants

Further quantum knot invariants for 10_157.

A1 Invariants.

Weight	Invariant
1	$2q^{-3}-2q^{-5}+3q^{-7}-q^{-9}+q^{-11}+q^{-13}-2q^{-15}+2q^{-17}-3q^{-19}+q^{-21}$
2	$q^{-4}+3q^{-6}-5q^{-8}-3q^{-10}+14q^{-12}-4q^{-14}-14q^{-16}+18q^{-18}+3q^{-20}-18q^{-22}+10q^{-24}+8q^{-26}-11q^{-28}-2q^{-30}+8q^{-32}+2q^{-34}-14q^{-36}+4q^{-38}+14q^{-40}-18q^{-42}-3q^{-44}+19q^{-46}-9q^{-48}-7q^{-50}+10q^{-52}-q^{-54}-3q^{-56}+q^{-58}$
3	$2q^{-5}+2q^{-7}-12q^{-11}-4q^{-13}+20q^{-15}+25q^{-17}-19q^{-19}-51q^{-21}+9q^{-23}+75q^{-25}+19q^{-27}-86q^{-29}-54q^{-31}+86q^{-33}+82q^{-35}-68q^{-37}-100q^{-39}+41q^{-41}+105q^{-43}-16q^{-45}-96q^{-47}-5q^{-49}+79q^{-51}+25q^{-53}-59q^{-55}-46q^{-57}+34q^{-59}+59q^{-61}-9q^{-63}-79q^{-65}-19q^{-67}+88q^{-69}+56q^{-71}-89q^{-73}-81q^{-75}+73q^{-77}+101q^{-79}-45q^{-81}-105q^{-83}+15q^{-85}+91q^{-87}+10q^{-89}-62q^{-91}-22q^{-93}+34q^{-95}+21q^{-97}-19q^{-99}-11q^{-101}+5q^{-103}+7q^{-105}-q^{-107}-3q^{-109}+q^{-111}$
5	$2q^{-3}+2q^{-5}+4q^{-7}-12q^{-11}-24q^{-13}-14q^{-15}+6q^{-17}+54q^{-19}+98q^{-21}+54q^{-23}-66q^{-25}-197q^{-27}-237q^{-29}-79q^{-31}+273q^{-33}+535q^{-35}+423q^{-37}-103q^{-39}-775q^{-41}-1014q^{-43}-438q^{-45}+748q^{-47}+1636q^{-49}+1350q^{-51}-186q^{-53}-1966q^{-55}-2462q^{-57}-942q^{-59}+1726q^{-61}+3388q^{-63}+2420q^{-65}-764q^{-67}-3756q^{-69}-3916q^{-71}-748q^{-73}+3407q^{-75}+4987q^{-77}+2429q^{-79}-2355q^{-81}-5384q^{-83}-3920q^{-85}+933q^{-87}+5085q^{-89}+4872q^{-91}+496q^{-93}-4226q^{-95}-5202q^{-97}-1663q^{-99}+3129q^{-101}+4966q^{-103}+2386q^{-105}-2053q^{-107}-4345q^{-109}-2691q^{-111}+1141q^{-113}+3596q^{-115}+2688q^{-117}-467q^{-119}-2879q^{-121}-2543q^{-123}-35q^{-125}+2286q^{-127}+2442q^{-129}+473q^{-131}-1827q^{-133}-2490q^{-135}-974q^{-137}+1424q^{-139}+2684q^{-141}+1676q^{-143}-928q^{-145}-2999q^{-147}-2587q^{-149}+235q^{-151}+3207q^{-153}+3657q^{-155}+825q^{-157}-3155q^{-159}-4695q^{-161}-2126q^{-163}+2590q^{-165}+5395q^{-167}+3579q^{-169}-1510q^{-171}-5506q^{-173}-4812q^{-175}+32q^{-177}+4847q^{-179}+5499q^{-181}+1509q^{-183}-3539q^{-185}-5406q^{-187}-2730q^{-189}+1925q^{-191}+4553q^{-193}+3288q^{-195}-396q^{-197}-3228q^{-199}-3148q^{-201}-663q^{-203}+1861q^{-205}+2472q^{-207}+1112q^{-209}-761q^{-211}-1605q^{-213}-1085q^{-215}+103q^{-217}+886q^{-219}+774q^{-221}+145q^{-223}-371q^{-225}-453q^{-227}-193q^{-229}+135q^{-231}+231q^{-233}+109q^{-235}-27q^{-237}-92q^{-239}-64q^{-241}-2q^{-243}+42q^{-245}+30q^{-247}-2q^{-249}-13q^{-251}-9q^{-253}-2q^{-255}+2q^{-257}+7q^{-259}-q^{-261}-3q^{-263}+q^{-265}$
6	$1+3q^{-2}+3q^{-4}+3q^{-6}-3q^{-8}-11q^{-10}-27q^{-12}-33q^{-14}-9q^{-16}+43q^{-18}+99q^{-20}+129q^{-22}+113q^{-24}-45q^{-26}-267q^{-28}-445q^{-30}-381q^{-32}-53q^{-34}+477q^{-36}+1073q^{-38}+1130q^{-40}+478q^{-42}-828q^{-44}-2046q^{-46}-2515q^{-48}-1544q^{-50}+1000q^{-52}+3654q^{-54}+4885q^{-56}+3239q^{-58}-749q^{-60}-5699q^{-62}-8412q^{-64}-6194q^{-66}+354q^{-68}+8364q^{-70}+12685q^{-72}+10360q^{-74}+548q^{-76}-11562q^{-78}-18333q^{-80}-14889q^{-82}-1383q^{-84}+14847q^{-86}+24778q^{-88}+19812q^{-90}+1786q^{-92}-19340q^{-94}-30753q^{-96}-23892q^{-98}-1614q^{-100}+24502q^{-102}+36410q^{-104}+26299q^{-106}-1001q^{-108}-29247q^{-110}-40339q^{-112}-26570q^{-114}+5472q^{-116}+34040q^{-118}+41714q^{-120}+22840q^{-122}-10634q^{-124}-37368q^{-126}-40159q^{-128}-16268q^{-130}+16700q^{-132}+38262q^{-134}+34261q^{-136}+8499q^{-138}-21677q^{-140}-36303q^{-142}-25761q^{-144}+101q^{-146}+24307q^{-148}+30561q^{-150}+16556q^{-152}-7303q^{-154}-24199q^{-156}-23017q^{-158}-7265q^{-160}+12008q^{-162}+21086q^{-164}+15416q^{-166}-312q^{-168}-14278q^{-170}-17026q^{-172}-8260q^{-174}+5758q^{-176}+14712q^{-178}+13350q^{-180}+2510q^{-182}-9742q^{-184}-15244q^{-186}-10312q^{-188}+2144q^{-190}+13543q^{-192}+16584q^{-194}+8064q^{-196}-6636q^{-198}-18576q^{-200}-18720q^{-202}-5820q^{-204}+12496q^{-206}+24642q^{-208}+20997q^{-210}+2705q^{-212}-20211q^{-214}-31317q^{-216}-22258q^{-218}+2772q^{-220}+28476q^{-222}+37046q^{-224}+21702q^{-226}-10072q^{-228}-36442q^{-230}-40322q^{-232}-17887q^{-234}+17147q^{-236}+42003q^{-238}+40479q^{-240}+12046q^{-242}-23262q^{-244}-43806q^{-246}-36344q^{-248}-6373q^{-250}+26589q^{-252}+41873q^{-254}+29729q^{-256}+1035q^{-258}-26503q^{-260}-35741q^{-262}-22871q^{-264}+2567q^{-266}+23853q^{-268}+27849q^{-270}+15756q^{-272}-4144q^{-274}-18656q^{-276}-20412q^{-278}-9881q^{-280}+4539q^{-282}+13189q^{-284}+13297q^{-286}+5691q^{-288}-3444q^{-290}-8886q^{-292}-7864q^{-294}-2695q^{-296}+2329q^{-298}+5135q^{-300}+4255q^{-302}+1325q^{-304}-1704q^{-306}-2686q^{-308}-1904q^{-310}-528q^{-312}+898q^{-314}+1300q^{-316}+892q^{-318}-2q^{-320}-452q^{-322}-478q^{-324}-334q^{-326}+26q^{-328}+217q^{-330}+227q^{-332}+39q^{-334}-39q^{-336}-64q^{-338}-74q^{-340}-12q^{-342}+26q^{-344}+44q^{-346}-2q^{-348}-4q^{-350}-q^{-352}-12q^{-354}-2q^{-356}+2q^{-358}+7q^{-360}-q^{-362}-3q^{-364}+q^{-366}$

A2 Invariants.

Weight	Invariant
1,0	$2q^{-6}-q^{-8}+2q^{-10}+3q^{-16}-q^{-18}+2q^{-20}-2q^{-22}-q^{-24}-2q^{-28}+q^{-30}$
1,1	$2q^{-10}-4q^{-14}+28q^{-16}-60q^{-18}+118q^{-20}-190q^{-22}+272q^{-24}-340q^{-26}+385q^{-28}-374q^{-30}+316q^{-32}-196q^{-34}+37q^{-36}+150q^{-38}-346q^{-40}+502q^{-42}-635q^{-44}+696q^{-46}-698q^{-48}+630q^{-50}-493q^{-52}+324q^{-54}-130q^{-56}-56q^{-58}+206q^{-60}-314q^{-62}+358q^{-64}-354q^{-66}+314q^{-68}-248q^{-70}+184q^{-72}-120q^{-74}+69q^{-76}-38q^{-78}+18q^{-80}-6q^{-82}+q^{-84}$
2,0	$q^{-10}+4q^{-12}-2q^{-14}-4q^{-16}+5q^{-18}+4q^{-20}-4q^{-22}-2q^{-24}+7q^{-26}+6q^{-28}-6q^{-30}+2q^{-32}+8q^{-34}-4q^{-36}-3q^{-38}+4q^{-40}-q^{-42}-6q^{-44}+2q^{-48}-7q^{-50}-5q^{-52}+5q^{-54}+2q^{-56}-9q^{-58}+2q^{-60}+7q^{-62}-q^{-66}+4q^{-70}-2q^{-72}-2q^{-74}+q^{-76}$

A3 Invariants.

Weight	Invariant
0,1,0	$3q^{-12}-3q^{-14}+2q^{-16}+9q^{-18}-9q^{-20}+4q^{-22}+13q^{-24}-13q^{-26}+4q^{-28}+11q^{-30}-9q^{-32}-2q^{-34}+5q^{-36}-3q^{-38}-5q^{-40}-4q^{-42}+6q^{-44}-2q^{-46}-11q^{-48}+13q^{-50}-13q^{-54}+12q^{-56}+q^{-58}-8q^{-60}+7q^{-62}-3q^{-66}+q^{-68}$
1,0,0	$2q^{-9}-q^{-11}+3q^{-13}-q^{-15}+2q^{-17}+2q^{-21}+q^{-23}+q^{-27}-2q^{-29}-3q^{-33}+q^{-35}-2q^{-37}+q^{-39}$
1,0,1	$2q^{-16}+q^{-18}-5q^{-20}+20q^{-22}-10q^{-24}-12q^{-26}+80q^{-28}-122q^{-30}+123q^{-32}-23q^{-34}-142q^{-36}+301q^{-38}-363q^{-40}+274q^{-42}-35q^{-44}-229q^{-46}+427q^{-48}-447q^{-50}+281q^{-52}-60q^{-54}-153q^{-56}+180q^{-58}-114q^{-60}-4q^{-62}+38q^{-64}+73q^{-66}-226q^{-68}+349q^{-70}-294q^{-72}+85q^{-74}+185q^{-76}-396q^{-78}+423q^{-80}-286q^{-82}+48q^{-84}+170q^{-86}-267q^{-88}+235q^{-90}-108q^{-92}-23q^{-94}+94q^{-96}-91q^{-98}+46q^{-100}-3q^{-102}-17q^{-104}+15q^{-106}-6q^{-108}+q^{-110}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$3q^{-18}-2q^{-20}+q^{-22}+7q^{-24}-3q^{-28}+11q^{-30}+6q^{-32}-7q^{-34}+3q^{-36}+12q^{-38}-q^{-40}-11q^{-42}+7q^{-44}+8q^{-46}-14q^{-48}-5q^{-50}+11q^{-52}-11q^{-54}-13q^{-56}+8q^{-58}-q^{-60}-12q^{-62}+2q^{-64}+11q^{-66}-3q^{-68}-6q^{-70}+9q^{-72}+6q^{-74}-8q^{-76}-q^{-78}+6q^{-80}-2q^{-82}-2q^{-84}+q^{-86}$
1,0,0,0	$2q^{-12}-q^{-14}+3q^{-16}+q^{-20}+2q^{-22}+2q^{-26}+2q^{-30}-q^{-32}+q^{-34}-2q^{-36}-2q^{-40}-2q^{-42}+q^{-44}-2q^{-46}+q^{-48}$

B2 Invariants.

Weight	Invariant
0,1	$3q^{-12}-5q^{-14}+10q^{-16}-11q^{-18}+15q^{-20}-14q^{-22}+13q^{-24}-9q^{-26}+4q^{-28}+3q^{-30}-11q^{-32}+18q^{-34}-23q^{-36}+27q^{-38}-27q^{-40}+24q^{-42}-18q^{-44}+12q^{-46}-5q^{-48}-3q^{-50}+8q^{-52}-13q^{-54}+14q^{-56}-15q^{-58}+12q^{-60}-9q^{-62}+6q^{-64}-3q^{-66}+q^{-68}$
1,0	$3q^{-18}+q^{-20}-4q^{-22}-4q^{-24}+6q^{-26}+10q^{-28}-13q^{-32}-5q^{-34}+14q^{-36}+13q^{-38}-7q^{-40}-14q^{-42}+q^{-44}+15q^{-46}+5q^{-48}-11q^{-50}-7q^{-52}+8q^{-54}+7q^{-56}-6q^{-58}-10q^{-60}+2q^{-62}+10q^{-64}-q^{-66}-12q^{-68}-3q^{-70}+10q^{-72}+5q^{-74}-11q^{-76}-10q^{-78}+9q^{-80}+14q^{-82}-4q^{-84}-16q^{-86}-3q^{-88}+13q^{-90}+10q^{-92}-7q^{-94}-10q^{-96}+q^{-98}+8q^{-100}+3q^{-102}-3q^{-104}-3q^{-106}+q^{-110}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$3q^{-18}-4q^{-20}+7q^{-22}-5q^{-24}+12q^{-26}-10q^{-28}+13q^{-30}-9q^{-32}+13q^{-34}-7q^{-36}+4q^{-38}-q^{-40}+q^{-42}+8q^{-44}-13q^{-46}+13q^{-48}-17q^{-50}+19q^{-52}-23q^{-54}+16q^{-56}-22q^{-58}+17q^{-60}-13q^{-62}+8q^{-64}-8q^{-66}+3q^{-68}+5q^{-70}-6q^{-72}+7q^{-74}-11q^{-76}+13q^{-78}-10q^{-80}+10q^{-82}-10q^{-84}+8q^{-86}-4q^{-88}+3q^{-90}-3q^{-92}+q^{-94}$

G2 Invariants.

Weight

Invariant

1,0

$q^{-28}-2q^{-32}+12q^{-34}-18q^{-36}+19q^{-38}-9q^{-40}-10q^{-42}+42q^{-44}-60q^{-46}+66q^{-48}-44q^{-50}-4q^{-52}+56q^{-54}-96q^{-56}+101q^{-58}-67q^{-60}+7q^{-62}+49q^{-64}-79q^{-66}+74q^{-68}-31q^{-70}-19q^{-72}+61q^{-74}-66q^{-76}+37q^{-78}+18q^{-80}-70q^{-82}+105q^{-84}-99q^{-86}+61q^{-88}+4q^{-90}-72q^{-92}+119q^{-94}-134q^{-96}+99q^{-98}-38q^{-100}-37q^{-102}+85q^{-104}-102q^{-106}+73q^{-108}-17q^{-110}-38q^{-112}+64q^{-114}-58q^{-116}+13q^{-118}+43q^{-120}-81q^{-122}+85q^{-124}-51q^{-126}+52q^{-130}-83q^{-132}+85q^{-134}-59q^{-136}+20q^{-138}+14q^{-140}-40q^{-142}+45q^{-144}-34q^{-146}+22q^{-148}-5q^{-150}-4q^{-152}+8q^{-154}-10q^{-156}+6q^{-158}-3q^{-160}+q^{-162}$

.

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["10 157"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+6t^{2}-11t+13-11t^{-1}+6t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{6}+4z^{2}+1$

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]=

$\{7,t+1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 49, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{8}a^{-6}+z^{8}a^{-8}+z^{7}a^{-5}+5z^{7}a^{-7}+4z^{7}a^{-9}+2z^{6}a^{-6}+8z^{6}a^{-8}+6z^{6}a^{-10}+z^{5}a^{-5}-3z^{5}a^{-7}+4z^{5}a^{-11}+3z^{4}a^{-4}-3z^{4}a^{-6}-15z^{4}a^{-8}-8z^{4}a^{-10}+z^{4}a^{-12}-2z^{3}a^{-5}-6z^{3}a^{-7}-8z^{3}a^{-9}-4z^{3}a^{-11}-5z^{2}a^{-4}+7z^{2}a^{-8}+2z^{2}a^{-10}+4za^{-7}+4za^{-9}+2a^{-4}-a^{-8}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {...}

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

Vassiliev invariants

V2 and V3:

(4, 8)

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 $16$ $64$ $128$ ${\frac {1016}{3}}$ ${\frac {184}{3}}$ $1024$ ${\frac {5824}{3}}$ ${\frac {1120}{3}}$ $288$ ${\frac {2048}{3}}$ $2048$ ${\frac {16256}{3}}$ ${\frac {2944}{3}}$ ${\frac {168062}{15}}$ ${\frac {584}{5}}$ ${\frac {218888}{45}}$ ${\frac {610}{9}}$ ${\frac {10142}{15}}$

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 $t^{r}q^{j}$ 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 10 157. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ 0 1 2 3 4 5 6 7 8 χ 21                 1 1 19               3   -3 17             3 1   2 15           5 3     -2 13         4 3       1 11       4 5         1 9     3 4           -1 7   1 4             3 5 1 3               -2 3 2                 2

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=3$ $i=5$ $r=0$ ${\mathbb {Z} }^{2}$ ${\mathbb {Z} }$ $r=1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=4$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{28}-4q^{27}+2q^{26}+12q^{25}-21q^{24}+40q^{22}-43q^{21}-15q^{20}+72q^{19}-53q^{18}-33q^{17}+88q^{16}-47q^{15}-43q^{14}+79q^{13}-28q^{12}-41q^{11}+51q^{10}-7q^{9}-26q^{8}+19q^{7}+3q^{6}-8q^{5}+2q^{4}+q^{3}$
3	$q^{54}-4q^{53}+2q^{52}+8q^{51}-q^{50}-20q^{49}-6q^{48}+48q^{47}+12q^{46}-76q^{45}-46q^{44}+120q^{43}+93q^{42}-152q^{41}-166q^{40}+180q^{39}+239q^{38}-180q^{37}-320q^{36}+172q^{35}+384q^{34}-148q^{33}-427q^{32}+112q^{31}+454q^{30}-80q^{29}-452q^{28}+32q^{27}+441q^{26}+4q^{25}-398q^{24}-52q^{23}+350q^{22}+84q^{21}-277q^{20}-116q^{19}+209q^{18}+116q^{17}-127q^{16}-112q^{15}+69q^{14}+84q^{13}-22q^{12}-56q^{11}+3q^{10}+24q^{9}+10q^{8}-12q^{7}-2q^{6}+2q^{4}$
4	$q^{88}-4q^{87}+2q^{86}+8q^{85}-5q^{84}-26q^{82}+12q^{81}+49q^{80}-7q^{79}-8q^{78}-128q^{77}+10q^{76}+190q^{75}+77q^{74}-6q^{73}-427q^{72}-144q^{71}+418q^{70}+425q^{69}+212q^{68}-910q^{67}-676q^{66}+470q^{65}+1014q^{64}+867q^{63}-1256q^{62}-1492q^{61}+94q^{60}+1491q^{59}+1803q^{58}-1204q^{57}-2184q^{56}-555q^{55}+1597q^{54}+2587q^{53}-857q^{52}-2482q^{51}-1143q^{50}+1399q^{49}+2981q^{48}-434q^{47}-2416q^{46}-1523q^{45}+1034q^{44}+3000q^{43}+8q^{42}-2064q^{41}-1729q^{40}+527q^{39}+2696q^{38}+468q^{37}-1448q^{36}-1728q^{35}-79q^{34}+2050q^{33}+815q^{32}-651q^{31}-1408q^{30}-569q^{29}+1159q^{28}+822q^{27}+35q^{26}-798q^{25}-666q^{24}+357q^{23}+482q^{22}+301q^{21}-224q^{20}-397q^{19}-25q^{18}+118q^{17}+193q^{16}+24q^{15}-107q^{14}-51q^{13}-16q^{12}+42q^{11}+26q^{10}-5q^{9}-6q^{8}-8q^{7}+2q^{5}+q^{4}$
5	$q^{130}-4q^{129}+2q^{128}+8q^{127}-5q^{126}-4q^{125}-6q^{124}-8q^{123}+13q^{122}+40q^{121}+7q^{120}-48q^{119}-68q^{118}-36q^{117}+78q^{116}+176q^{115}+129q^{114}-144q^{113}-396q^{112}-296q^{111}+160q^{110}+692q^{109}+758q^{108}-32q^{107}-1179q^{106}-1484q^{105}-360q^{104}+1536q^{103}+2631q^{102}+1328q^{101}-1790q^{100}-4008q^{99}-2845q^{98}+1456q^{97}+5463q^{96}+5012q^{95}-525q^{94}-6636q^{93}-7500q^{92}-1220q^{91}+7330q^{90}+10060q^{89}+3465q^{88}-7288q^{87}-12315q^{86}-6064q^{85}+6636q^{84}+14056q^{83}+8554q^{82}-5472q^{81}-15120q^{80}-10780q^{79}+4067q^{78}+15596q^{77}+12534q^{76}-2640q^{75}-15570q^{74}-13752q^{73}+1245q^{72}+15184q^{71}+14605q^{70}-36q^{69}-14562q^{68}-15012q^{67}-1153q^{66}+13668q^{65}+15268q^{64}+2264q^{63}-12593q^{62}-15168q^{61}-3474q^{60}+11160q^{59}+14932q^{58}+4676q^{57}-9438q^{56}-14260q^{55}-5929q^{54}+7328q^{53}+13278q^{52}+6968q^{51}-4999q^{50}-11680q^{49}-7766q^{48}+2536q^{47}+9739q^{46}+7944q^{45}-277q^{44}-7304q^{43}-7553q^{42}-1616q^{41}+4886q^{40}+6480q^{39}+2752q^{38}-2520q^{37}-4995q^{36}-3196q^{35}+731q^{34}+3312q^{33}+2912q^{32}+472q^{31}-1811q^{30}-2228q^{29}-931q^{28}+644q^{27}+1392q^{26}+968q^{25}-31q^{24}-692q^{23}-645q^{22}-244q^{21}+206q^{20}+392q^{19}+208q^{18}-20q^{17}-119q^{16}-132q^{15}-56q^{14}+40q^{13}+50q^{12}+20q^{11}+12q^{10}-12q^{9}-12q^{8}-4q^{7}+2q^{6}+2q^{4}$
6	$q^{180}-4q^{179}+2q^{178}+8q^{177}-5q^{176}-4q^{175}-10q^{174}+12q^{173}-7q^{172}+4q^{171}+54q^{170}-23q^{169}-42q^{168}-72q^{167}+22q^{166}+18q^{165}+82q^{164}+242q^{163}-33q^{162}-233q^{161}-432q^{160}-122q^{159}+44q^{158}+532q^{157}+1136q^{156}+375q^{155}-635q^{154}-1858q^{153}-1498q^{152}-738q^{151}+1514q^{150}+4165q^{149}+3305q^{148}+245q^{147}-4664q^{146}-6522q^{145}-5907q^{144}+492q^{143}+9607q^{142}+12440q^{141}+7851q^{140}-4772q^{139}-15172q^{138}-20327q^{137}-10039q^{136}+11363q^{135}+26952q^{134}+27751q^{133}+7321q^{132}-19168q^{131}-41613q^{130}-35477q^{129}-1507q^{128}+36190q^{127}+55289q^{126}+36015q^{125}-7024q^{124}-56897q^{123}-68439q^{122}-31478q^{121}+28728q^{120}+75833q^{119}+71323q^{118}+21409q^{117}-55373q^{116}-93295q^{115}-66512q^{114}+6293q^{113}+79713q^{112}+97693q^{111}+53183q^{110}-40029q^{109}-101865q^{108}-92216q^{107}-18737q^{106}+70654q^{105}+108799q^{104}+76099q^{103}-21737q^{102}-98220q^{101}-104362q^{100}-37053q^{99}+57754q^{98}+108943q^{97}+88039q^{96}-7037q^{95}-89700q^{94}-107403q^{93}-48452q^{92}+45298q^{91}+104011q^{90}+93541q^{89}+5215q^{88}-78860q^{87}-106041q^{86}-57406q^{85}+31280q^{84}+95245q^{83}+96289q^{82}+19181q^{81}-63132q^{80}-100371q^{79}-66484q^{78}+12007q^{77}+79493q^{76}+95107q^{75}+36077q^{74}-39273q^{73}-86366q^{72}-72738q^{71}-12199q^{70}+53631q^{69}+84565q^{68}+50703q^{67}-9097q^{66}-60604q^{65}-68737q^{64}-33761q^{63}+20663q^{62}+60674q^{61}+53494q^{60}+17637q^{59}-27130q^{58}-49863q^{57}-41435q^{56}-7905q^{55}+28632q^{54}+39725q^{53}+28729q^{52}+1116q^{51}-22563q^{50}-31324q^{49}-19813q^{48}+2516q^{47}+17447q^{46}+21868q^{45}+12529q^{44}-1437q^{43}-13298q^{42}-14847q^{41}-7415q^{40}+1217q^{39}+8362q^{38}+9085q^{37}+5334q^{36}-1188q^{35}-5035q^{34}-5090q^{33}-3104q^{32}+352q^{31}+2537q^{30}+3116q^{29}+1525q^{28}-85q^{27}-1102q^{26}-1458q^{25}-879q^{24}-117q^{23}+572q^{22}+554q^{21}+384q^{20}+116q^{19}-152q^{18}-227q^{17}-174q^{16}-24q^{15}+24q^{14}+56q^{13}+52q^{12}+26q^{11}-5q^{10}-16q^{9}-8q^{8}-6q^{7}+2q^{4}+q^{3}$
7	Not Available

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
In[2]:=	PD[Knot[10, 157]]
Out[2]=	PD[X[1, 6, 2, 7], X[10, 4, 11, 3], X[16, 11, 17, 12], X[7, 15, 8, 14], X[15, 9, 16, 8], X[13, 1, 14, 20], X[19, 13, 20, 12], X[18, 6, 19, 5], X[2, 10, 3, 9], X[4, 18, 5, 17]]
In[3]:=	GaussCode[Knot[10, 157]]
Out[3]=	GaussCode[-1, -9, 2, -10, 8, 1, -4, 5, 9, -2, 3, 7, -6, 4, -5, -3, 10, -8, -7, 6]
In[4]:=	DTCode[Knot[10, 157]]
Out[4]=	DTCode[6, -10, -18, 14, -2, -16, 20, 8, -4, 12]
In[5]:=	br = BR[Knot[10, 157]]
Out[5]=	BR[3, {1, 1, 1, 2, 2, -1, 2, -1, 2, 2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 10}
In[7]:=	BraidIndex[Knot[10, 157]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[10, 157]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[10, 157]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, NotAvailable, 2}
In[10]:=	alex = Alexander[Knot[10, 157]][t]
Out[10]=	-3 6 11 2 3 13 - t + -- - -- - 11 t + 6 t - t 2 t t
In[11]:=	Conway[Knot[10, 157]][z]
Out[11]=	2 6 1 + 4 z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 157]}
In[13]:=	{KnotDet[Knot[10, 157]], KnotSignature[Knot[10, 157]]}
Out[13]=	{49, 4}
In[14]:=	Jones[Knot[10, 157]][q]
Out[14]=	2 3 4 5 6 7 8 9 10 2 q - 4 q + 7 q - 8 q + 9 q - 8 q + 6 q - 4 q + q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 157]}
In[16]:=	A2Invariant[Knot[10, 157]][q]
Out[16]=	6 8 10 16 18 20 22 24 28 30 2 q - q + 2 q + 3 q - q + 2 q - 2 q - q - 2 q + q
In[17]:=	HOMFLYPT[Knot[10, 157]][a, z]
Out[17]=	2 2 2 4 4 4 6 -8 2 z 2 z 5 z z 3 z 2 z z -a + -- + -- - ---- + ---- + -- - ---- + ---- - -- 4 8 6 4 8 6 4 6 a a a a a a a a
In[18]:=	Kauffman[Knot[10, 157]][a, z]
Out[18]=	2 2 2 3 3 3 -8 2 4 z 4 z 2 z 7 z 5 z 4 z 8 z 6 z -a + -- + --- + --- + ---- + ---- - ---- - ---- - ---- - ---- - 4 9 7 10 8 4 11 9 7 a a a a a a a a a 3 4 4 4 4 4 5 5 5 6 2 z z 8 z 15 z 3 z 3 z 4 z 3 z z 6 z ---- + --- - ---- - ----- - ---- + ---- + ---- - ---- + -- + ---- + 5 12 10 8 6 4 11 7 5 10 a a a a a a a a a a 6 6 7 7 7 8 8 8 z 2 z 4 z 5 z z z z ---- + ---- + ---- + ---- + -- + -- + -- 8 6 9 7 5 8 6 a a a a a a a
In[19]:=	{Vassiliev[2][Knot[10, 157]], Vassiliev[3][Knot[10, 157]]}
Out[19]=	{4, 8}
In[20]:=	Kh[Knot[10, 157]][q, t]
Out[20]=	3 5 5 7 7 2 9 2 9 3 11 3 2 q + q + 3 q t + q t + 4 q t + 3 q t + 4 q t + 4 q t + 11 4 13 4 13 5 15 5 15 6 17 6 5 q t + 4 q t + 3 q t + 5 q t + 3 q t + 3 q t + 17 7 19 7 21 8 q t + 3 q t + q t
In[21]:=	ColouredJones[Knot[10, 157], 2][q]
Out[21]=	3 4 5 6 7 8 9 10 11 q + 2 q - 8 q + 3 q + 19 q - 26 q - 7 q + 51 q - 41 q - 12 13 14 15 16 17 18 28 q + 79 q - 43 q - 47 q + 88 q - 33 q - 53 q + 19 20 21 22 24 25 26 27 72 q - 15 q - 43 q + 40 q - 21 q + 12 q + 2 q - 4 q + 28 q

See/edit the Rolfsen_Splice_Template.