8 5

8_4

8_6

Visit 8 5's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 8 5's page at Knotilus! Visit 8 5's page at the original Knot Atlas! 8 5 is also known as the pretzel knot P(3,3,2).

Symmetric alternative representation

Pretzel P(3,3,2) form Photo 01-09-2017 besalu.jpg.

Sum of 8.5 ; church of Besalu, Catalogna

Knot presentations

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

Minimum Braid Representative: Length is 8, 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 $\{4,6\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [1][-11] Hyperbolic Volume 6.99719 A-Polynomial See Data:8 5/A-polynomial

[edit Notes for 8 5'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 8 5's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+3t^{2}-4t+5-4t^{-1}+3t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}-3z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 21, 4 }
Jones polynomial	$q^{8}-2q^{7}+3q^{6}-4q^{5}+3q^{4}-3q^{3}+3q^{2}-q+1$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-4}+z^{4}a^{-2}-5z^{4}a^{-4}+z^{4}a^{-6}+4z^{2}a^{-2}-8z^{2}a^{-4}+3z^{2}a^{-6}+4a^{-2}-5a^{-4}+2a^{-6}$
Kauffman polynomial (db, data sources)	$z^{7}a^{-3}+z^{7}a^{-5}+z^{6}a^{-2}+4z^{6}a^{-4}+3z^{6}a^{-6}-3z^{5}a^{-3}+z^{5}a^{-5}+4z^{5}a^{-7}-5z^{4}a^{-2}-15z^{4}a^{-4}-7z^{4}a^{-6}+3z^{4}a^{-8}-10z^{3}a^{-5}-8z^{3}a^{-7}+2z^{3}a^{-9}+8z^{2}a^{-2}+15z^{2}a^{-4}+4z^{2}a^{-6}-2z^{2}a^{-8}+z^{2}a^{-10}+3za^{-3}+7za^{-5}+4za^{-7}-4a^{-2}-5a^{-4}-2a^{-6}$
The A2 invariant	$1+q^{-2}+2q^{-4}+2q^{-6}-3q^{-12}-q^{-14}-q^{-16}+q^{-20}+q^{-24}$
The G2 invariant	$q^{-2}+3q^{-6}-2q^{-8}+2q^{-10}+q^{-12}-2q^{-14}+7q^{-16}-5q^{-18}+5q^{-20}+q^{-22}-2q^{-24}+8q^{-26}-5q^{-28}+5q^{-30}+2q^{-32}-2q^{-34}+3q^{-36}-3q^{-38}+2q^{-42}-4q^{-44}+2q^{-46}-4q^{-48}-2q^{-50}+2q^{-52}-9q^{-54}+4q^{-56}-7q^{-58}+q^{-62}-7q^{-64}+7q^{-66}-7q^{-68}+4q^{-70}+2q^{-72}-5q^{-74}+5q^{-76}-2q^{-78}+q^{-80}+4q^{-82}-2q^{-84}+3q^{-86}+q^{-88}-q^{-90}+4q^{-92}-4q^{-94}+3q^{-96}-2q^{-100}+3q^{-102}-3q^{-104}+3q^{-106}-q^{-108}+q^{-110}-3q^{-114}+2q^{-116}-2q^{-118}+2q^{-120}-q^{-122}-q^{-128}+q^{-130}-q^{-132}+q^{-134}$

Further Quantum Invariants

Further quantum knot invariants for 8_5.

A1 Invariants.

Weight	Invariant
1	$q+2q^{-3}-q^{-9}-q^{-11}+q^{-13}-q^{-15}+q^{-17}$
2	$q^{6}-q^{2}+2+2q^{-2}-2q^{-4}+q^{-6}+2q^{-8}-2q^{-10}-q^{-12}+2q^{-14}-q^{-16}-2q^{-18}+q^{-20}+q^{-22}-q^{-24}+2q^{-28}-2q^{-32}+2q^{-34}+q^{-36}-3q^{-38}+q^{-40}-q^{-44}+q^{-46}$
3	$q^{15}-q^{11}-q^{9}+2q^{7}+3q^{5}-4q-q^{-1}+4q^{-3}+4q^{-5}-3q^{-7}-4q^{-9}+5q^{-13}+2q^{-15}-4q^{-17}-2q^{-19}+3q^{-21}+4q^{-23}-2q^{-25}-4q^{-27}+5q^{-31}-q^{-33}-4q^{-35}-q^{-37}+6q^{-39}+q^{-41}-5q^{-43}-3q^{-45}+3q^{-47}+3q^{-49}-q^{-51}-3q^{-53}-2q^{-55}+4q^{-57}+4q^{-59}-2q^{-61}-6q^{-63}+q^{-65}+5q^{-67}-3q^{-71}+q^{-73}+q^{-75}-q^{-79}-q^{-85}+q^{-87}$
4	$q^{28}-q^{24}-q^{22}-q^{20}+3q^{18}+3q^{16}+q^{14}-2q^{12}-7q^{10}-q^{8}+4q^{6}+8q^{4}+5q^{2}-7-8q^{-2}-6q^{-4}+5q^{-6}+13q^{-8}+4q^{-10}-3q^{-12}-13q^{-14}-7q^{-16}+9q^{-18}+12q^{-20}+10q^{-22}-7q^{-24}-14q^{-26}-4q^{-28}+5q^{-30}+15q^{-32}+4q^{-34}-11q^{-36}-12q^{-38}-2q^{-40}+12q^{-42}+8q^{-44}-5q^{-46}-12q^{-48}-5q^{-50}+10q^{-52}+9q^{-54}-4q^{-56}-12q^{-58}-3q^{-60}+11q^{-62}+10q^{-64}-4q^{-66}-12q^{-68}-3q^{-70}+8q^{-72}+13q^{-74}+2q^{-76}-9q^{-78}-9q^{-80}-6q^{-82}+7q^{-84}+13q^{-86}+5q^{-88}-7q^{-90}-19q^{-92}-6q^{-94}+14q^{-96}+17q^{-98}+5q^{-100}-19q^{-102}-15q^{-104}+4q^{-106}+14q^{-108}+10q^{-110}-8q^{-112}-8q^{-114}-q^{-116}+3q^{-118}+6q^{-120}-3q^{-122}-2q^{-124}+q^{-126}+2q^{-130}-2q^{-132}-q^{-138}+q^{-140}$
5	$q^{45}-q^{41}-q^{39}-q^{37}+3q^{33}+4q^{31}+q^{29}-2q^{27}-5q^{25}-7q^{23}-2q^{21}+6q^{19}+11q^{17}+9q^{15}+q^{13}-11q^{11}-16q^{9}-11q^{7}+3q^{5}+17q^{3}+21q+10q^{-1}-8q^{-3}-23q^{-5}-25q^{-7}-6q^{-9}+18q^{-11}+30q^{-13}+25q^{-15}+q^{-17}-27q^{-19}-35q^{-21}-16q^{-23}+14q^{-25}+36q^{-27}+36q^{-29}+7q^{-31}-28q^{-33}-43q^{-35}-26q^{-37}+10q^{-39}+41q^{-41}+40q^{-43}+6q^{-45}-34q^{-47}-46q^{-49}-24q^{-51}+17q^{-53}+45q^{-55}+33q^{-57}-7q^{-59}-38q^{-61}-39q^{-63}-5q^{-65}+33q^{-67}+39q^{-69}+10q^{-71}-23q^{-73}-32q^{-75}-11q^{-77}+23q^{-79}+29q^{-81}+6q^{-83}-18q^{-85}-24q^{-87}-5q^{-89}+23q^{-91}+24q^{-93}-2q^{-95}-25q^{-97}-25q^{-99}-2q^{-101}+26q^{-103}+29q^{-105}+6q^{-107}-23q^{-109}-33q^{-111}-18q^{-113}+8q^{-115}+30q^{-117}+33q^{-119}+11q^{-121}-20q^{-123}-40q^{-125}-34q^{-127}+42q^{-131}+55q^{-133}+23q^{-135}-28q^{-137}-63q^{-139}-45q^{-141}+12q^{-143}+58q^{-145}+57q^{-147}+6q^{-149}-47q^{-151}-54q^{-153}-19q^{-155}+28q^{-157}+45q^{-159}+22q^{-161}-17q^{-163}-31q^{-165}-16q^{-167}+6q^{-169}+19q^{-171}+12q^{-173}-4q^{-175}-11q^{-177}-4q^{-179}+3q^{-181}+4q^{-183}+q^{-185}-q^{-187}-2q^{-189}+2q^{-193}+q^{-195}-2q^{-197}-q^{-203}+q^{-205}$
6	$q^{66}-q^{62}-q^{60}-q^{58}+4q^{52}+4q^{50}+q^{48}-2q^{46}-5q^{44}-6q^{42}-8q^{40}+q^{38}+8q^{36}+13q^{34}+12q^{32}+6q^{30}-5q^{28}-22q^{26}-21q^{24}-15q^{22}+2q^{20}+19q^{18}+33q^{16}+32q^{14}+7q^{12}-16q^{10}-40q^{8}-43q^{6}-30q^{4}+8q^{2}+44+56q^{-2}+50q^{-4}+10q^{-6}-34q^{-8}-73q^{-10}-67q^{-12}-28q^{-14}+23q^{-16}+76q^{-18}+88q^{-20}+59q^{-22}-13q^{-24}-72q^{-26}-98q^{-28}-80q^{-30}-11q^{-32}+68q^{-34}+117q^{-36}+94q^{-38}+29q^{-40}-57q^{-42}-123q^{-44}-119q^{-46}-44q^{-48}+60q^{-50}+121q^{-52}+125q^{-54}+56q^{-56}-54q^{-58}-137q^{-60}-135q^{-62}-52q^{-64}+53q^{-66}+136q^{-68}+138q^{-70}+53q^{-72}-69q^{-74}-141q^{-76}-122q^{-78}-36q^{-80}+79q^{-82}+144q^{-84}+114q^{-86}+8q^{-88}-93q^{-90}-129q^{-92}-86q^{-94}+16q^{-96}+102q^{-98}+114q^{-100}+45q^{-102}-42q^{-104}-98q^{-106}-85q^{-108}-11q^{-110}+63q^{-112}+83q^{-114}+36q^{-116}-24q^{-118}-64q^{-120}-48q^{-122}+5q^{-124}+51q^{-126}+52q^{-128}-q^{-130}-47q^{-132}-60q^{-134}-20q^{-136}+37q^{-138}+74q^{-140}+58q^{-142}-11q^{-144}-68q^{-146}-84q^{-148}-42q^{-150}+22q^{-152}+81q^{-154}+95q^{-156}+49q^{-158}-20q^{-160}-83q^{-162}-101q^{-164}-75q^{-166}-2q^{-168}+85q^{-170}+131q^{-172}+106q^{-174}+15q^{-176}-90q^{-178}-161q^{-180}-143q^{-182}-29q^{-184}+114q^{-186}+188q^{-188}+151q^{-190}+26q^{-192}-127q^{-194}-205q^{-196}-149q^{-198}+q^{-200}+138q^{-202}+186q^{-204}+124q^{-206}-20q^{-208}-142q^{-210}-156q^{-212}-67q^{-214}+47q^{-216}+114q^{-218}+109q^{-220}+26q^{-222}-61q^{-224}-86q^{-226}-50q^{-228}+7q^{-230}+44q^{-232}+52q^{-234}+16q^{-236}-24q^{-238}-30q^{-240}-14q^{-242}+3q^{-244}+11q^{-246}+15q^{-248}+3q^{-250}-9q^{-252}-5q^{-254}+2q^{-258}+3q^{-262}-q^{-264}-3q^{-266}+2q^{-268}+q^{-270}+q^{-272}-2q^{-274}-q^{-280}+q^{-282}$

A2 Invariants.

Weight	Invariant
1,0	$1+q^{-2}+2q^{-4}+2q^{-6}-3q^{-12}-q^{-14}-q^{-16}+q^{-20}+q^{-24}$
1,1	$q^{4}+6-4q^{-2}+12q^{-4}-10q^{-6}+16q^{-8}-12q^{-10}+10q^{-12}-8q^{-14}-2q^{-16}+2q^{-18}-14q^{-20}+12q^{-22}-22q^{-24}+22q^{-26}-20q^{-28}+22q^{-30}-12q^{-32}+16q^{-34}-q^{-36}+4q^{-40}-10q^{-42}+5q^{-44}-10q^{-46}+6q^{-48}-6q^{-50}+5q^{-52}-2q^{-54}+4q^{-56}-4q^{-58}+3q^{-60}-2q^{-62}+2q^{-64}-2q^{-66}+q^{-68}$
2,0	$q^{4}+q^{2}+1+q^{-2}+3q^{-4}+3q^{-6}+2q^{-8}+q^{-12}-2q^{-14}-4q^{-16}-4q^{-18}-3q^{-20}-3q^{-22}-q^{-24}+2q^{-26}+3q^{-28}+3q^{-30}+3q^{-32}+3q^{-34}-q^{-36}+q^{-40}-2q^{-44}-q^{-46}-q^{-48}-q^{-50}-q^{-52}+q^{-56}+q^{-60}$
3,0	$q^{12}+q^{10}+q^{8}+2q^{2}+4+4q^{-2}+q^{-4}-q^{-6}+3q^{-10}+3q^{-12}-2q^{-14}-8q^{-16}-6q^{-18}-3q^{-20}+q^{-22}-4q^{-24}-5q^{-26}-3q^{-28}+5q^{-30}+7q^{-32}+6q^{-34}+4q^{-36}+6q^{-38}+8q^{-40}+5q^{-42}+q^{-44}-3q^{-46}-2q^{-48}-6q^{-50}-7q^{-52}-8q^{-54}-q^{-56}-2q^{-58}-2q^{-60}-3q^{-62}+2q^{-64}+6q^{-66}+2q^{-68}-2q^{-70}-3q^{-72}+4q^{-74}+7q^{-76}+3q^{-78}-3q^{-80}-3q^{-82}+q^{-84}+4q^{-86}+q^{-88}-2q^{-90}-3q^{-92}-q^{-96}-q^{-98}-q^{-100}+q^{-104}+q^{-108}$

A3 Invariants.

Weight	Invariant
0,1,0	$1+3q^{-4}+3q^{-6}+3q^{-8}+5q^{-10}+3q^{-12}-q^{-14}-q^{-16}-6q^{-18}-8q^{-20}-4q^{-22}-3q^{-24}-q^{-26}+3q^{-28}+5q^{-30}+4q^{-32}+2q^{-34}+q^{-36}+q^{-38}-3q^{-40}-q^{-42}+q^{-44}-2q^{-46}+2q^{-50}-q^{-52}-q^{-54}+q^{-56}$
1,0,0	$q^{-1}+q^{-3}+3q^{-5}+2q^{-7}+3q^{-9}-q^{-13}-3q^{-15}-3q^{-17}-2q^{-19}-q^{-21}+q^{-23}+2q^{-27}+q^{-31}$
1,0,1	$q^{2}+6q^{-2}+2q^{-4}+11q^{-6}+8q^{-8}+6q^{-10}+12q^{-12}-7q^{-14}+7q^{-16}-14q^{-18}-8q^{-20}-11q^{-22}-19q^{-24}-3q^{-26}-13q^{-28}+7q^{-30}-3q^{-32}+13q^{-34}+7q^{-36}+11q^{-38}+9q^{-40}+q^{-42}+10q^{-44}-8q^{-46}+7q^{-48}-8q^{-50}-2q^{-52}-q^{-54}-9q^{-56}+4q^{-58}-6q^{-60}+q^{-62}+2q^{-64}-2q^{-66}+2q^{-68}+q^{-72}+2q^{-74}-q^{-76}-q^{-78}+2q^{-80}-2q^{-82}+q^{-84}+q^{-86}-2q^{-88}+q^{-90}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{-2}+q^{-4}+3q^{-6}+5q^{-8}+6q^{-10}+7q^{-12}+8q^{-14}+5q^{-16}+2q^{-18}-2q^{-20}-7q^{-22}-13q^{-24}-14q^{-26}-12q^{-28}-9q^{-30}-6q^{-32}+4q^{-34}+10q^{-36}+9q^{-38}+11q^{-40}+10q^{-42}+4q^{-44}-q^{-46}-q^{-48}-4q^{-50}-5q^{-52}-3q^{-54}-q^{-58}-q^{-60}+2q^{-62}+q^{-64}-q^{-66}+q^{-70}$
1,0,0,0	$q^{-2}+q^{-4}+3q^{-6}+3q^{-8}+3q^{-10}+3q^{-12}-q^{-16}-4q^{-18}-3q^{-20}-4q^{-22}-2q^{-24}-q^{-26}+q^{-28}+q^{-30}+q^{-32}+2q^{-34}+q^{-38}$

B2 Invariants.

Weight	Invariant
0,1	$1+3q^{-4}-q^{-6}+3q^{-8}-q^{-10}+3q^{-12}-q^{-14}+q^{-16}-2q^{-20}+2q^{-22}-5q^{-24}+3q^{-26}-5q^{-28}+3q^{-30}-4q^{-32}+2q^{-34}-q^{-36}+q^{-38}+q^{-40}-q^{-42}+3q^{-44}-2q^{-46}+2q^{-48}-2q^{-50}+q^{-52}-q^{-54}+q^{-56}$
1,0	$q^{2}+3q^{-6}+2q^{-8}+3q^{-14}+3q^{-16}+2q^{-18}-2q^{-20}-q^{-22}+q^{-24}+q^{-26}-3q^{-28}-5q^{-30}-3q^{-32}-q^{-34}-q^{-36}-3q^{-38}-q^{-40}+q^{-42}+2q^{-44}+q^{-48}+2q^{-50}+4q^{-52}+q^{-54}-q^{-56}+2q^{-60}+q^{-62}-2q^{-64}-2q^{-66}+2q^{-70}-q^{-72}-2q^{-74}-q^{-76}+q^{-78}+2q^{-80}-q^{-84}-q^{-86}+q^{-90}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{-2}+3q^{-6}+q^{-8}+6q^{-10}+3q^{-12}+6q^{-14}+3q^{-16}+5q^{-18}-q^{-20}-2q^{-22}-5q^{-24}-7q^{-26}-6q^{-28}-9q^{-30}-3q^{-32}-6q^{-34}+3q^{-36}-q^{-38}+7q^{-40}+2q^{-42}+7q^{-44}+q^{-46}+4q^{-48}-q^{-54}-2q^{-56}-2q^{-60}+2q^{-62}-2q^{-64}+q^{-66}-q^{-68}+2q^{-70}-q^{-72}-q^{-76}+q^{-78}$

G2 Invariants.

Weight

Invariant

1,0

$q^{-2}+3q^{-6}-2q^{-8}+2q^{-10}+q^{-12}-2q^{-14}+7q^{-16}-5q^{-18}+5q^{-20}+q^{-22}-2q^{-24}+8q^{-26}-5q^{-28}+5q^{-30}+2q^{-32}-2q^{-34}+3q^{-36}-3q^{-38}+2q^{-42}-4q^{-44}+2q^{-46}-4q^{-48}-2q^{-50}+2q^{-52}-9q^{-54}+4q^{-56}-7q^{-58}+q^{-62}-7q^{-64}+7q^{-66}-7q^{-68}+4q^{-70}+2q^{-72}-5q^{-74}+5q^{-76}-2q^{-78}+q^{-80}+4q^{-82}-2q^{-84}+3q^{-86}+q^{-88}-q^{-90}+4q^{-92}-4q^{-94}+3q^{-96}-2q^{-100}+3q^{-102}-3q^{-104}+3q^{-106}-q^{-108}+q^{-110}-3q^{-114}+2q^{-116}-2q^{-118}+2q^{-120}-q^{-122}-q^{-128}+q^{-130}-q^{-132}+q^{-134}$

.

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["8 5"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+3t^{2}-4t+5-4t^{-1}+3t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{6}-3z^{4}-z^{2}+1$

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]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 21, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_141, ...}

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

Vassiliev invariants

V2 and V3:

(-1, -3)

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 $-4$ $-24$ $8$ $-{\frac {62}{3}}$ ${\frac {86}{3}}$ $96$ $208$ $96$ $104$ $-{\frac {32}{3}}$ $288$ ${\frac {248}{3}}$ $-{\frac {344}{3}}$ ${\frac {31409}{30}}$ ${\frac {834}{5}}$ ${\frac {16618}{45}}$ ${\frac {2095}{18}}$ $-{\frac {1231}{30}}$

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 8 5. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -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         1 2       -1 7       2 2         0 5     1 1           0 3   1 3             2 1                   0 -1 1                 1

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{22}-2q^{21}+q^{20}+2q^{19}-6q^{18}+5q^{17}+3q^{16}-10q^{15}+7q^{14}+5q^{13}-12q^{12}+6q^{11}+7q^{10}-12q^{9}+3q^{8}+8q^{7}-9q^{6}+7q^{4}-5q^{3}-q^{2}+4q-1-q^{-1}+q^{-2}$
3	$q^{42}-2q^{41}+q^{40}-q^{37}+2q^{36}-4q^{34}+2q^{33}+7q^{32}-4q^{31}-11q^{30}+6q^{29}+13q^{28}-4q^{27}-17q^{26}+5q^{25}+15q^{24}-17q^{22}-q^{21}+13q^{20}+6q^{19}-12q^{18}-8q^{17}+10q^{16}+9q^{15}-6q^{14}-13q^{13}+6q^{12}+11q^{11}-14q^{9}+q^{8}+9q^{7}+6q^{6}-11q^{5}-4q^{4}+5q^{3}+7q^{2}-4q-4+4q^{-2}-q^{-4}-q^{-5}+q^{-6}$
4	$q^{68}-2q^{67}+q^{66}-2q^{64}+5q^{63}-4q^{62}+2q^{61}-3q^{60}-3q^{59}+14q^{58}-7q^{57}-2q^{56}-10q^{55}-3q^{54}+32q^{53}-3q^{52}-12q^{51}-29q^{50}-7q^{49}+56q^{48}+9q^{47}-15q^{46}-49q^{45}-20q^{44}+68q^{43}+21q^{42}-7q^{41}-55q^{40}-33q^{39}+65q^{38}+21q^{37}+4q^{36}-44q^{35}-38q^{34}+54q^{33}+12q^{32}+12q^{31}-30q^{30}-37q^{29}+40q^{28}+3q^{27}+20q^{26}-17q^{25}-36q^{24}+25q^{23}-4q^{22}+27q^{21}-4q^{20}-32q^{19}+11q^{18}-14q^{17}+28q^{16}+11q^{15}-21q^{14}+q^{13}-23q^{12}+18q^{11}+18q^{10}-4q^{9}+3q^{8}-26q^{7}+2q^{6}+12q^{5}+6q^{4}+10q^{3}-17q^{2}-6q+1+4q^{-1}+11q^{-2}-5q^{-3}-3q^{-4}-3q^{-5}-q^{-6}+5q^{-7}-q^{-10}-q^{-11}+q^{-12}$
5	$q^{100}-2q^{99}+q^{98}-2q^{96}+3q^{95}+2q^{94}-4q^{93}-q^{92}+q^{91}+6q^{89}+q^{88}-11q^{87}-8q^{86}+8q^{85}+16q^{84}+13q^{83}-12q^{82}-33q^{81}-23q^{80}+22q^{79}+55q^{78}+36q^{77}-29q^{76}-80q^{75}-58q^{74}+29q^{73}+108q^{72}+87q^{71}-28q^{70}-126q^{69}-115q^{68}+11q^{67}+143q^{66}+138q^{65}+4q^{64}-139q^{63}-157q^{62}-23q^{61}+137q^{60}+158q^{59}+35q^{58}-117q^{57}-160q^{56}-45q^{55}+111q^{54}+143q^{53}+45q^{52}-88q^{51}-137q^{50}-48q^{49}+83q^{48}+120q^{47}+45q^{46}-65q^{45}-111q^{44}-49q^{43}+55q^{42}+101q^{41}+51q^{40}-41q^{39}-88q^{38}-55q^{37}+21q^{36}+80q^{35}+60q^{34}-8q^{33}-59q^{32}-61q^{31}-17q^{30}+46q^{29}+61q^{28}+23q^{27}-19q^{26}-49q^{25}-45q^{24}+5q^{23}+39q^{22}+35q^{21}+21q^{20}-15q^{19}-44q^{18}-26q^{17}+3q^{16}+18q^{15}+36q^{14}+20q^{13}-15q^{12}-26q^{11}-19q^{10}-12q^{9}+17q^{8}+28q^{7}+13q^{6}-2q^{5}-14q^{4}-24q^{3}-7q^{2}+9q+15+13q^{-1}+4q^{-2}-13q^{-3}-11q^{-4}-5q^{-5}+q^{-6}+8q^{-7}+9q^{-8}-q^{-9}-3q^{-10}-3q^{-11}-4q^{-12}+4q^{-14}+q^{-15}-q^{-18}-q^{-19}+q^{-20}$
6	$q^{138}-2q^{137}+q^{136}-2q^{134}+3q^{133}+2q^{131}-7q^{130}+3q^{129}+4q^{128}-5q^{127}+5q^{126}-2q^{125}-3q^{124}-11q^{123}+15q^{122}+16q^{121}-9q^{120}-3q^{119}-19q^{118}-19q^{117}-5q^{116}+55q^{115}+52q^{114}-17q^{113}-40q^{112}-76q^{111}-55q^{110}+20q^{109}+142q^{108}+135q^{107}-12q^{106}-107q^{105}-190q^{104}-144q^{103}+34q^{102}+264q^{101}+279q^{100}+50q^{99}-155q^{98}-327q^{97}-294q^{96}-22q^{95}+342q^{94}+432q^{93}+175q^{92}-118q^{91}-401q^{90}-437q^{89}-136q^{88}+324q^{87}+503q^{86}+280q^{85}-27q^{84}-376q^{83}-483q^{82}-223q^{81}+251q^{80}+477q^{79}+298q^{78}+36q^{77}-307q^{76}-437q^{75}-237q^{74}+192q^{73}+413q^{72}+256q^{71}+52q^{70}-250q^{69}-368q^{68}-221q^{67}+155q^{66}+356q^{65}+216q^{64}+65q^{63}-204q^{62}-315q^{61}-222q^{60}+109q^{59}+303q^{58}+200q^{57}+105q^{56}-144q^{55}-268q^{54}-242q^{53}+35q^{52}+229q^{51}+187q^{50}+161q^{49}-57q^{48}-199q^{47}-254q^{46}-51q^{45}+127q^{44}+144q^{43}+197q^{42}+44q^{41}-93q^{40}-224q^{39}-116q^{38}+12q^{37}+58q^{36}+178q^{35}+116q^{34}+29q^{33}-139q^{32}-118q^{31}-71q^{30}-47q^{29}+95q^{28}+114q^{27}+112q^{26}-29q^{25}-49q^{24}-75q^{23}-108q^{22}-9q^{21}+39q^{20}+108q^{19}+37q^{18}+37q^{17}-10q^{16}-85q^{15}-58q^{14}-40q^{13}+39q^{12}+19q^{11}+63q^{10}+49q^{9}-13q^{8}-29q^{7}-52q^{6}-14q^{5}-32q^{4}+24q^{3}+43q^{2}+26q+15-12q^{-1}-8q^{-2}-44q^{-3}-12q^{-4}+5q^{-5}+13q^{-6}+18q^{-7}+12q^{-8}+15q^{-9}-19q^{-10}-11q^{-11}-9q^{-12}-4q^{-13}+q^{-14}+6q^{-15}+14q^{-16}-2q^{-17}-3q^{-19}-3q^{-20}-4q^{-21}-q^{-22}+5q^{-23}+q^{-25}-q^{-28}-q^{-29}+q^{-30}$
7	$q^{182}-2q^{181}+q^{180}-2q^{178}+3q^{177}-q^{174}-3q^{173}+6q^{172}-q^{171}-6q^{170}+5q^{169}-3q^{168}+2q^{166}+q^{165}+13q^{164}-4q^{163}-19q^{162}-7q^{161}-9q^{160}+14q^{159}+29q^{158}+18q^{157}+17q^{156}-31q^{155}-66q^{154}-47q^{153}-11q^{152}+77q^{151}+119q^{150}+77q^{149}+4q^{148}-131q^{147}-203q^{146}-146q^{145}+226q^{143}+332q^{142}+233q^{141}+2q^{140}-328q^{139}-488q^{138}-374q^{137}-40q^{136}+427q^{135}+679q^{134}+562q^{133}+119q^{132}-503q^{131}-868q^{130}-766q^{129}-256q^{128}+507q^{127}+1020q^{126}+996q^{125}+439q^{124}-466q^{123}-1113q^{122}-1177q^{121}-631q^{120}+349q^{119}+1131q^{118}+1308q^{117}+807q^{116}-211q^{115}-1082q^{114}-1363q^{113}-937q^{112}+85q^{111}+999q^{110}+1339q^{109}+988q^{108}+24q^{107}-888q^{106}-1286q^{105}-999q^{104}-69q^{103}+815q^{102}+1191q^{101}+942q^{100}+91q^{99}-736q^{98}-1108q^{97}-900q^{96}-79q^{95}+707q^{94}+1031q^{93}+828q^{92}+68q^{91}-655q^{90}-969q^{89}-803q^{88}-71q^{87}+627q^{86}+920q^{85}+770q^{84}+94q^{83}-564q^{82}-865q^{81}-772q^{80}-144q^{79}+495q^{78}+808q^{77}+772q^{76}+207q^{75}-399q^{74}-734q^{73}-770q^{72}-288q^{71}+280q^{70}+648q^{69}+767q^{68}+367q^{67}-163q^{66}-533q^{65}-728q^{64}-445q^{63}+16q^{62}+406q^{61}+688q^{60}+494q^{59}+106q^{58}-252q^{57}-590q^{56}-527q^{55}-239q^{54}+100q^{53}+483q^{52}+507q^{51}+320q^{50}+65q^{49}-326q^{48}-454q^{47}-384q^{46}-200q^{45}+179q^{44}+346q^{43}+371q^{42}+304q^{41}-3q^{40}-214q^{39}-332q^{38}-351q^{37}-109q^{36}+66q^{35}+209q^{34}+336q^{33}+212q^{32}+71q^{31}-101q^{30}-271q^{29}-213q^{28}-158q^{27}-43q^{26}+153q^{25}+191q^{24}+202q^{23}+126q^{22}-49q^{21}-99q^{20}-168q^{19}-176q^{18}-54q^{17}+10q^{16}+112q^{15}+162q^{14}+91q^{13}+66q^{12}-23q^{11}-108q^{10}-94q^{9}-104q^{8}-39q^{7}+43q^{6}+52q^{5}+98q^{4}+73q^{3}+13q^{2}-5q-60-70q^{-1}-41q^{-2}-36q^{-3}+22q^{-4}+46q^{-5}+35q^{-6}+47q^{-7}+15q^{-8}-11q^{-9}-21q^{-10}-46q^{-11}-23q^{-12}-4q^{-13}-2q^{-14}+21q^{-15}+22q^{-16}+18q^{-17}+14q^{-18}-13q^{-19}-12q^{-20}-8q^{-21}-13q^{-22}-4q^{-23}+q^{-24}+7q^{-25}+13q^{-26}+q^{-27}+q^{-29}-4q^{-30}-3q^{-31}-4q^{-32}-q^{-33}+4q^{-34}+q^{-35}+q^{-37}-q^{-40}-q^{-41}+q^{-42}$

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[8, 5]]
Out[2]=	PD[X[6, 2, 7, 1], X[8, 4, 9, 3], X[2, 8, 3, 7], X[14, 10, 15, 9], X[12, 5, 13, 6], X[4, 13, 5, 14], X[16, 12, 1, 11], X[10, 16, 11, 15]]
In[3]:=	GaussCode[Knot[8, 5]]
Out[3]=	GaussCode[1, -3, 2, -6, 5, -1, 3, -2, 4, -8, 7, -5, 6, -4, 8, -7]
In[4]:=	DTCode[Knot[8, 5]]
Out[4]=	DTCode[6, 8, 12, 2, 14, 16, 4, 10]
In[5]:=	br = BR[Knot[8, 5]]
Out[5]=	BR[3, {1, 1, 1, -2, 1, 1, 1, -2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 8}
In[7]:=	BraidIndex[Knot[8, 5]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[8, 5]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[8, 5]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, {4, 6}, 1}
In[10]:=	alex = Alexander[Knot[8, 5]][t]
Out[10]=	-3 3 4 2 3 5 - t + -- - - - 4 t + 3 t - t 2 t t
In[11]:=	Conway[Knot[8, 5]][z]
Out[11]=	2 4 6 1 - z - 3 z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[8, 5], Knot[10, 141]}
In[13]:=	{KnotDet[Knot[8, 5]], KnotSignature[Knot[8, 5]]}
Out[13]=	{21, 4}
In[14]:=	Jones[Knot[8, 5]][q]
Out[14]=	2 3 4 5 6 7 8 1 - q + 3 q - 3 q + 3 q - 4 q + 3 q - 2 q + q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[8, 5]}
In[16]:=	A2Invariant[Knot[8, 5]][q]
Out[16]=	2 4 6 12 14 16 20 24 1 + q + 2 q + 2 q - 3 q - q - q + q + q
In[17]:=	HOMFLYPT[Knot[8, 5]][a, z]
Out[17]=	2 2 2 4 4 4 6 2 5 4 3 z 8 z 4 z z 5 z z z -- - -- + -- + ---- - ---- + ---- + -- - ---- + -- - -- 6 4 2 6 4 2 6 4 2 4 a a a a a a a a a a
In[18]:=	Kauffman[Knot[8, 5]][a, z]
Out[18]=	2 2 2 2 2 -2 5 4 4 z 7 z 3 z z 2 z 4 z 15 z 8 z -- - -- - -- + --- + --- + --- + --- - ---- + ---- + ----- + ---- + 6 4 2 7 5 3 10 8 6 4 2 a a a a a a a a a a a 3 3 3 4 4 4 4 5 5 5 2 z 8 z 10 z 3 z 7 z 15 z 5 z 4 z z 3 z ---- - ---- - ----- + ---- - ---- - ----- - ---- + ---- + -- - ---- + 9 7 5 8 6 4 2 7 5 3 a a a a a a a a a a 6 6 6 7 7 3 z 4 z z z z ---- + ---- + -- + -- + -- 6 4 2 5 3 a a a a a
In[19]:=	{Vassiliev[2][Knot[8, 5]], Vassiliev[3][Knot[8, 5]]}
Out[19]=	{-1, -3}
In[20]:=	Kh[Knot[8, 5]][q, t]
Out[20]=	3 3 5 1 q 5 7 7 2 9 2 9 3 3 q + q + ---- + -- + q t + 2 q t + 2 q t + q t + 2 q t + 2 t q t 11 3 11 4 13 4 13 5 15 5 17 6 2 q t + q t + 2 q t + q t + q t + q t
In[21]:=	ColouredJones[Knot[8, 5], 2][q]
Out[21]=	-2 1 2 3 4 6 7 8 9 -1 + q - - + 4 q - q - 5 q + 7 q - 9 q + 8 q + 3 q - 12 q + q 10 11 12 13 14 15 16 17 7 q + 6 q - 12 q + 5 q + 7 q - 10 q + 3 q + 5 q - 18 19 20 21 22 6 q + 2 q + q - 2 q + q

See/edit the Rolfsen_Splice_Template.