9 26

9_25

9_27

Visit 9 26's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 26's page at Knotilus! Visit 9 26's page at the original Knot Atlas! 9 26 Quick Notes

9 26 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X5,12,6,13 X3,11,4,10 X11,3,12,2 X13,18,14,1 X7,15,8,14 X17,7,18,6 X9,17,10,16 X15,9,16,8
Gauss code	-1, 4, -3, 1, -2, 7, -6, 9, -8, 3, -4, 2, -5, 6, -9, 8, -7, 5
Dowker-Thistlethwaite code	4 10 12 14 16 2 18 8 6
Conway Notation	[311112]

Minimum Braid Representative: Length is 9, width is 4. Braid index is 4.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 3 Bridge index 2 Super bridge index $\{4,6\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-2][-9] Hyperbolic Volume 10.5958 A-Polynomial See Data:9 26/A-polynomial

[edit Notes for 9 26's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $3$ Rasmussen s-Invariant 2

[edit Notes for 9 26's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-5t^{2}+11t-13+11t^{-1}-5t^{-2}+t^{-3}$
Conway polynomial	$z^{6}+z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 47, 2 }
Jones polynomial	$q^{7}-3q^{6}+5q^{5}-7q^{4}+8q^{3}-8q^{2}+7q-4+3q^{-1}-q^{-2}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+4z^{4}a^{-2}-2z^{4}a^{-4}-z^{4}+6z^{2}a^{-2}-5z^{2}a^{-4}+z^{2}a^{-6}-2z^{2}+3a^{-2}-3a^{-4}+a^{-6}$
Kauffman polynomial (db, data sources)	$z^{8}a^{-2}+z^{8}a^{-4}+3z^{7}a^{-1}+6z^{7}a^{-3}+3z^{7}a^{-5}+5z^{6}a^{-2}+6z^{6}a^{-4}+4z^{6}a^{-6}+3z^{6}+az^{5}-6z^{5}a^{-1}-11z^{5}a^{-3}-z^{5}a^{-5}+3z^{5}a^{-7}-16z^{4}a^{-2}-14z^{4}a^{-4}-5z^{4}a^{-6}+z^{4}a^{-8}-8z^{4}-2az^{3}+3z^{3}a^{-1}+7z^{3}a^{-3}-2z^{3}a^{-5}-4z^{3}a^{-7}+13z^{2}a^{-2}+11z^{2}a^{-4}+2z^{2}a^{-6}-z^{2}a^{-8}+5z^{2}-za^{-1}-za^{-3}+za^{-5}+za^{-7}-3a^{-2}-3a^{-4}-a^{-6}$
The A2 invariant	$-q^{6}+q^{4}+1+3q^{-2}-q^{-4}+2q^{-6}-q^{-8}-2q^{-14}+q^{-16}-q^{-18}+q^{-22}$
The G2 invariant	$q^{32}-2q^{30}+4q^{28}-7q^{26}+5q^{24}-4q^{22}-4q^{20}+16q^{18}-23q^{16}+28q^{14}-23q^{12}+8q^{10}+15q^{8}-39q^{6}+53q^{4}-49q^{2}+30+2q^{-2}-31q^{-4}+51q^{-6}-49q^{-8}+35q^{-10}-5q^{-12}-23q^{-14}+35q^{-16}-28q^{-18}+6q^{-20}+25q^{-22}-40q^{-24}+44q^{-26}-23q^{-28}-10q^{-30}+46q^{-32}-73q^{-34}+76q^{-36}-53q^{-38}+12q^{-40}+33q^{-42}-67q^{-44}+78q^{-46}-62q^{-48}+28q^{-50}+5q^{-52}-38q^{-54}+44q^{-56}-31q^{-58}+4q^{-60}+21q^{-62}-33q^{-64}+26q^{-66}-4q^{-68}-25q^{-70}+44q^{-72}-50q^{-74}+40q^{-76}-15q^{-78}-15q^{-80}+38q^{-82}-46q^{-84}+44q^{-86}-27q^{-88}+9q^{-90}+8q^{-92}-21q^{-94}+24q^{-96}-19q^{-98}+13q^{-100}-4q^{-102}-2q^{-104}+4q^{-106}-6q^{-108}+4q^{-110}-2q^{-112}+q^{-114}$

Further Quantum Invariants

Further quantum knot invariants for 9_26.

A1 Invariants.

Weight	Invariant
1	$-q^{5}+2q^{3}-q+3q^{-1}-q^{-3}+q^{-7}-2q^{-9}+2q^{-11}-2q^{-13}+q^{-15}$
2	$q^{16}-2q^{14}-2q^{12}+6q^{10}-2q^{8}-7q^{6}+10q^{4}+2q^{2}-12+10q^{-2}+6q^{-4}-12q^{-6}+4q^{-8}+7q^{-10}-5q^{-12}-5q^{-14}+3q^{-16}+6q^{-18}-10q^{-20}-q^{-22}+14q^{-24}-9q^{-26}-5q^{-28}+12q^{-30}-5q^{-32}-5q^{-34}+6q^{-36}-q^{-38}-2q^{-40}+q^{-42}$
3	$-q^{33}+2q^{31}+2q^{29}-3q^{27}-6q^{25}+2q^{23}+13q^{21}-q^{19}-19q^{17}-6q^{15}+26q^{13}+16q^{11}-27q^{9}-31q^{7}+27q^{5}+42q^{3}-16q-51q^{-1}+9q^{-3}+56q^{-5}+4q^{-7}-54q^{-9}-15q^{-11}+50q^{-13}+19q^{-15}-38q^{-17}-28q^{-19}+25q^{-21}+30q^{-23}-8q^{-25}-32q^{-27}-9q^{-29}+30q^{-31}+29q^{-33}-27q^{-35}-43q^{-37}+20q^{-39}+54q^{-41}-10q^{-43}-57q^{-45}+53q^{-49}+9q^{-51}-45q^{-53}-14q^{-55}+32q^{-57}+14q^{-59}-21q^{-61}-12q^{-63}+13q^{-65}+10q^{-67}-8q^{-69}-5q^{-71}+3q^{-73}+3q^{-75}-q^{-77}-2q^{-79}+q^{-81}$
4	$q^{56}-2q^{54}-2q^{52}+3q^{50}+3q^{48}+6q^{46}-9q^{44}-13q^{42}+2q^{40}+10q^{38}+31q^{36}-9q^{34}-41q^{32}-23q^{30}+8q^{28}+81q^{26}+29q^{24}-55q^{22}-85q^{20}-53q^{18}+115q^{16}+119q^{14}+6q^{12}-130q^{10}-173q^{8}+63q^{6}+186q^{4}+138q^{2}-83-269q^{-2}-60q^{-4}+161q^{-6}+241q^{-8}+25q^{-10}-270q^{-12}-161q^{-14}+76q^{-16}+255q^{-18}+108q^{-20}-196q^{-22}-186q^{-24}-4q^{-26}+198q^{-28}+138q^{-30}-91q^{-32}-167q^{-34}-70q^{-36}+113q^{-38}+145q^{-40}+30q^{-42}-122q^{-44}-137q^{-46}-5q^{-48}+139q^{-50}+170q^{-52}-49q^{-54}-195q^{-56}-142q^{-58}+91q^{-60}+276q^{-62}+59q^{-64}-179q^{-66}-244q^{-68}-11q^{-70}+280q^{-72}+147q^{-74}-80q^{-76}-238q^{-78}-103q^{-80}+178q^{-82}+146q^{-84}+21q^{-86}-142q^{-88}-113q^{-90}+68q^{-92}+75q^{-94}+50q^{-96}-50q^{-98}-66q^{-100}+18q^{-102}+20q^{-104}+30q^{-106}-14q^{-108}-26q^{-110}+8q^{-112}+2q^{-114}+11q^{-116}-3q^{-118}-8q^{-120}+3q^{-122}+3q^{-126}-q^{-128}-2q^{-130}+q^{-132}$
5	$-q^{85}+2q^{83}+2q^{81}-3q^{79}-3q^{77}-3q^{75}+q^{73}+9q^{71}+13q^{69}-2q^{67}-20q^{65}-22q^{63}-7q^{61}+25q^{59}+49q^{57}+33q^{55}-33q^{53}-87q^{51}-70q^{49}+12q^{47}+118q^{45}+151q^{43}+44q^{41}-140q^{39}-239q^{37}-149q^{35}+90q^{33}+325q^{31}+318q^{29}+23q^{27}-350q^{25}-498q^{23}-242q^{21}+283q^{19}+651q^{17}+513q^{15}-83q^{13}-709q^{11}-812q^{9}-203q^{7}+644q^{5}+1030q^{3}+571q-444q^{-1}-1161q^{-3}-901q^{-5}+162q^{-7}+1138q^{-9}+1163q^{-11}+163q^{-13}-1012q^{-15}-1308q^{-17}-440q^{-19}+807q^{-21}+1315q^{-23}+646q^{-25}-568q^{-27}-1239q^{-29}-763q^{-31}+370q^{-33}+1078q^{-35}+786q^{-37}-173q^{-39}-913q^{-41}-771q^{-43}+43q^{-45}+737q^{-47}+723q^{-49}+97q^{-51}-572q^{-53}-695q^{-55}-228q^{-57}+398q^{-59}+683q^{-61}+398q^{-63}-215q^{-65}-672q^{-67}-599q^{-69}-17q^{-71}+649q^{-73}+829q^{-75}+281q^{-77}-581q^{-79}-1025q^{-81}-601q^{-83}+431q^{-85}+1174q^{-87}+912q^{-89}-205q^{-91}-1204q^{-93}-1173q^{-95}-90q^{-97}+1103q^{-99}+1337q^{-101}+391q^{-103}-880q^{-105}-1346q^{-107}-639q^{-109}+569q^{-111}+1213q^{-113}+788q^{-115}-255q^{-117}-967q^{-119}-799q^{-121}-8q^{-123}+667q^{-125}+701q^{-127}+174q^{-129}-392q^{-131}-536q^{-133}-233q^{-135}+181q^{-137}+354q^{-139}+215q^{-141}-45q^{-143}-209q^{-145}-164q^{-147}-4q^{-149}+104q^{-151}+98q^{-153}+24q^{-155}-43q^{-157}-59q^{-159}-16q^{-161}+22q^{-163}+23q^{-165}+8q^{-167}-7q^{-169}-10q^{-171}-6q^{-173}+5q^{-175}+7q^{-177}-2q^{-179}-3q^{-181}+3q^{-189}-q^{-191}-2q^{-193}+q^{-195}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{6}+q^{4}+1+3q^{-2}-q^{-4}+2q^{-6}-q^{-8}-2q^{-14}+q^{-16}-q^{-18}+q^{-22}$
1,1	$q^{20}-4q^{18}+10q^{16}-22q^{14}+42q^{12}-70q^{10}+100q^{8}-140q^{6}+177q^{4}-196q^{2}+208-188q^{-2}+157q^{-4}-86q^{-6}+2q^{-8}+94q^{-10}-193q^{-12}+280q^{-14}-354q^{-16}+392q^{-18}-402q^{-20}+372q^{-22}-312q^{-24}+228q^{-26}-133q^{-28}+34q^{-30}+58q^{-32}-124q^{-34}+170q^{-36}-190q^{-38}+194q^{-40}-176q^{-42}+146q^{-44}-118q^{-46}+86q^{-48}-58q^{-50}+36q^{-52}-20q^{-54}+10q^{-56}-4q^{-58}+q^{-60}$
2,0	$q^{18}-q^{16}-2q^{14}+q^{12}+2q^{10}-2q^{8}-4q^{6}+4q^{4}+6q^{2}-2-2q^{-2}+8q^{-4}+2q^{-6}-4q^{-8}+q^{-10}+5q^{-12}-2q^{-14}-4q^{-16}+2q^{-18}-2q^{-20}-7q^{-22}+q^{-24}+4q^{-26}-5q^{-28}-q^{-30}+7q^{-32}+3q^{-34}-4q^{-36}+5q^{-40}-4q^{-44}+q^{-48}-q^{-50}-q^{-52}+q^{-56}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{14}-2q^{12}+2q^{8}-6q^{6}+4q^{4}+6q^{2}-7+6q^{-2}+9q^{-4}-10q^{-6}+3q^{-8}+8q^{-10}-6q^{-12}-q^{-14}+3q^{-16}-5q^{-20}-4q^{-22}+7q^{-24}-4q^{-26}-7q^{-28}+12q^{-30}-q^{-32}-8q^{-34}+9q^{-36}-6q^{-40}+4q^{-42}-2q^{-46}+q^{-48}$
1,0,0	$-q^{7}+q^{5}-q^{3}+2q+3q^{-3}+2q^{-7}+q^{-9}-2q^{-15}-3q^{-19}+q^{-21}-q^{-23}+q^{-25}+q^{-29}$

A4 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	$-q^{14}+2q^{12}-4q^{10}+6q^{8}-8q^{6}+10q^{4}-10q^{2}+11-8q^{-2}+7q^{-4}-3q^{-8}+10q^{-10}-14q^{-12}+19q^{-14}-21q^{-16}+20q^{-18}-19q^{-20}+14q^{-22}-11q^{-24}+4q^{-26}+q^{-28}-6q^{-30}+9q^{-32}-10q^{-34}+11q^{-36}-10q^{-38}+8q^{-40}-6q^{-42}+4q^{-44}-2q^{-46}+q^{-48}$
1,0	$q^{24}-2q^{20}-2q^{18}+2q^{16}+4q^{14}-2q^{12}-7q^{10}-2q^{8}+9q^{6}+8q^{4}-4q^{2}-10+q^{-2}+12q^{-4}+8q^{-6}-8q^{-8}-9q^{-10}+3q^{-12}+10q^{-14}-8q^{-18}-2q^{-20}+7q^{-22}+3q^{-24}-7q^{-26}-5q^{-28}+5q^{-30}+6q^{-32}-5q^{-34}-8q^{-36}+2q^{-38}+9q^{-40}-10q^{-44}-4q^{-46}+10q^{-48}+9q^{-50}-5q^{-52}-11q^{-54}-q^{-56}+10q^{-58}+6q^{-60}-5q^{-62}-7q^{-64}+5q^{-68}+2q^{-70}-2q^{-72}-2q^{-74}+q^{-78}$

D4 Invariants.

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

G2 Invariants.

Weight

Invariant

1,0

$q^{32}-2q^{30}+4q^{28}-7q^{26}+5q^{24}-4q^{22}-4q^{20}+16q^{18}-23q^{16}+28q^{14}-23q^{12}+8q^{10}+15q^{8}-39q^{6}+53q^{4}-49q^{2}+30+2q^{-2}-31q^{-4}+51q^{-6}-49q^{-8}+35q^{-10}-5q^{-12}-23q^{-14}+35q^{-16}-28q^{-18}+6q^{-20}+25q^{-22}-40q^{-24}+44q^{-26}-23q^{-28}-10q^{-30}+46q^{-32}-73q^{-34}+76q^{-36}-53q^{-38}+12q^{-40}+33q^{-42}-67q^{-44}+78q^{-46}-62q^{-48}+28q^{-50}+5q^{-52}-38q^{-54}+44q^{-56}-31q^{-58}+4q^{-60}+21q^{-62}-33q^{-64}+26q^{-66}-4q^{-68}-25q^{-70}+44q^{-72}-50q^{-74}+40q^{-76}-15q^{-78}-15q^{-80}+38q^{-82}-46q^{-84}+44q^{-86}-27q^{-88}+9q^{-90}+8q^{-92}-21q^{-94}+24q^{-96}-19q^{-98}+13q^{-100}-4q^{-102}-2q^{-104}+4q^{-106}-6q^{-108}+4q^{-110}-2q^{-112}+q^{-114}$

.

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["9 26"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-5t^{2}+11t-13+11t^{-1}-5t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 47, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{8}a^{-2}+z^{8}a^{-4}+3z^{7}a^{-1}+6z^{7}a^{-3}+3z^{7}a^{-5}+5z^{6}a^{-2}+6z^{6}a^{-4}+4z^{6}a^{-6}+3z^{6}+az^{5}-6z^{5}a^{-1}-11z^{5}a^{-3}-z^{5}a^{-5}+3z^{5}a^{-7}-16z^{4}a^{-2}-14z^{4}a^{-4}-5z^{4}a^{-6}+z^{4}a^{-8}-8z^{4}-2az^{3}+3z^{3}a^{-1}+7z^{3}a^{-3}-2z^{3}a^{-5}-4z^{3}a^{-7}+13z^{2}a^{-2}+11z^{2}a^{-4}+2z^{2}a^{-6}-z^{2}a^{-8}+5z^{2}-za^{-1}-za^{-3}+za^{-5}+za^{-7}-3a^{-2}-3a^{-4}-a^{-6}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n25, ...}

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

Vassiliev invariants

V2 and V3:

(0, -1)

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 $0$ $-8$ $0$ $-32$ $-8$ $0$ $-{\frac {176}{3}}$ ${\frac {64}{3}}$ $-40$ $0$ $32$ $0$ $0$ $0$ ${\frac {248}{3}}$ $-{\frac {88}{3}}$ $-{\frac {112}{3}}$ $0$

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=$2 is the signature of 9 26. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{20}-3q^{19}+q^{18}+8q^{17}-14q^{16}+q^{15}+25q^{14}-31q^{13}-3q^{12}+48q^{11}-46q^{10}-12q^{9}+64q^{8}-49q^{7}-20q^{6}+64q^{5}-37q^{4}-23q^{3}+48q^{2}-19q-19+26q^{-1}-5q^{-2}-11q^{-3}+9q^{-4}-3q^{-6}+q^{-7}$
3	$q^{39}-3q^{38}+q^{37}+4q^{36}+q^{35}-11q^{34}-2q^{33}+22q^{32}+4q^{31}-36q^{30}-11q^{29}+57q^{28}+22q^{27}-82q^{26}-42q^{25}+111q^{24}+66q^{23}-135q^{22}-99q^{21}+158q^{20}+130q^{19}-169q^{18}-162q^{17}+174q^{16}+186q^{15}-168q^{14}-201q^{13}+151q^{12}+210q^{11}-130q^{10}-206q^{9}+98q^{8}+200q^{7}-73q^{6}-175q^{5}+33q^{4}+161q^{3}-15q^{2}-123q-14+101q^{-1}+20q^{-2}-65q^{-3}-29q^{-4}+43q^{-5}+24q^{-6}-22q^{-7}-19q^{-8}+11q^{-9}+11q^{-10}-4q^{-11}-5q^{-12}+3q^{-14}-q^{-15}$
4	$q^{64}-3q^{63}+q^{62}+4q^{61}-3q^{60}+4q^{59}-14q^{58}+6q^{57}+18q^{56}-12q^{55}+10q^{54}-48q^{53}+18q^{52}+62q^{51}-22q^{50}+8q^{49}-132q^{48}+34q^{47}+162q^{46}+3q^{45}+q^{44}-313q^{43}+5q^{42}+325q^{41}+128q^{40}+33q^{39}-594q^{38}-130q^{37}+483q^{36}+355q^{35}+166q^{34}-885q^{33}-363q^{32}+548q^{31}+593q^{30}+383q^{29}-1070q^{28}-596q^{27}+495q^{26}+739q^{25}+602q^{24}-1101q^{23}-740q^{22}+363q^{21}+754q^{20}+754q^{19}-986q^{18}-772q^{17}+180q^{16}+657q^{15}+830q^{14}-757q^{13}-712q^{12}-22q^{11}+475q^{10}+820q^{9}-453q^{8}-565q^{7}-201q^{6}+238q^{5}+711q^{4}-158q^{3}-349q^{2}-281q+17+502q^{-1}+28q^{-2}-128q^{-3}-233q^{-4}-106q^{-5}+266q^{-6}+71q^{-7}+8q^{-8}-120q^{-9}-110q^{-10}+98q^{-11}+39q^{-12}+38q^{-13}-36q^{-14}-58q^{-15}+25q^{-16}+8q^{-17}+20q^{-18}-4q^{-19}-18q^{-20}+4q^{-21}+5q^{-23}-3q^{-25}+q^{-26}$
5	$q^{95}-3q^{94}+q^{93}+4q^{92}-3q^{91}+q^{89}-6q^{88}+2q^{87}+13q^{86}-5q^{85}-11q^{84}-3q^{83}-3q^{82}+17q^{81}+28q^{80}-6q^{79}-49q^{78}-46q^{77}+13q^{76}+84q^{75}+102q^{74}-157q^{72}-206q^{71}-32q^{70}+248q^{69}+362q^{68}+139q^{67}-330q^{66}-620q^{65}-335q^{64}+392q^{63}+928q^{62}+666q^{61}-364q^{60}-1295q^{59}-1126q^{58}+224q^{57}+1640q^{56}+1709q^{55}+61q^{54}-1939q^{53}-2334q^{52}-483q^{51}+2106q^{50}+2980q^{49}+1007q^{48}-2173q^{47}-3527q^{46}-1566q^{45}+2075q^{44}+3979q^{43}+2124q^{42}-1911q^{41}-4270q^{40}-2598q^{39}+1651q^{38}+4423q^{37}+2986q^{36}-1363q^{35}-4450q^{34}-3264q^{33}+1069q^{32}+4350q^{31}+3443q^{30}-750q^{29}-4165q^{28}-3549q^{27}+443q^{26}+3883q^{25}+3566q^{24}-81q^{23}-3539q^{22}-3535q^{21}-251q^{20}+3069q^{19}+3424q^{18}+659q^{17}-2580q^{16}-3243q^{15}-959q^{14}+1936q^{13}+2948q^{12}+1330q^{11}-1366q^{10}-2574q^{9}-1467q^{8}+689q^{7}+2080q^{6}+1626q^{5}-191q^{4}-1574q^{3}-1492q^{2}-287q+1017+1366q^{-1}+526q^{-2}-559q^{-3}-1033q^{-4}-673q^{-5}+170q^{-6}+757q^{-7}+629q^{-8}+67q^{-9}-437q^{-10}-535q^{-11}-198q^{-12}+232q^{-13}+373q^{-14}+215q^{-15}-64q^{-16}-240q^{-17}-191q^{-18}-3q^{-19}+134q^{-20}+125q^{-21}+35q^{-22}-56q^{-23}-84q^{-24}-36q^{-25}+28q^{-26}+43q^{-27}+18q^{-28}-2q^{-29}-18q^{-30}-20q^{-31}+4q^{-32}+11q^{-33}+3q^{-34}-5q^{-37}+3q^{-39}-q^{-40}$
6	$q^{132}-3q^{131}+q^{130}+4q^{129}-3q^{128}-3q^{126}+9q^{125}-10q^{124}-3q^{123}+20q^{122}-15q^{121}-5q^{120}-8q^{119}+35q^{118}-16q^{117}-9q^{116}+48q^{115}-56q^{114}-41q^{113}-29q^{112}+120q^{111}+13q^{110}+21q^{109}+110q^{108}-185q^{107}-202q^{106}-160q^{105}+271q^{104}+184q^{103}+259q^{102}+359q^{101}-404q^{100}-692q^{99}-715q^{98}+260q^{97}+508q^{96}+1049q^{95}+1292q^{94}-352q^{93}-1560q^{92}-2228q^{91}-642q^{90}+477q^{89}+2494q^{88}+3640q^{87}+961q^{86}-2142q^{85}-4826q^{84}-3362q^{83}-1133q^{82}+3776q^{81}+7474q^{80}+4564q^{79}-1014q^{78}-7462q^{77}-7860q^{76}-5339q^{75}+3329q^{74}+11463q^{73}+10163q^{72}+2758q^{71}-8395q^{70}-12527q^{69}-11525q^{68}+327q^{67}+13706q^{66}+15848q^{65}+8319q^{64}-6842q^{63}-15408q^{62}-17574q^{61}-4251q^{60}+13448q^{59}+19636q^{58}+13584q^{57}-3723q^{56}-15842q^{55}-21632q^{54}-8541q^{53}+11510q^{52}+20960q^{51}+17032q^{50}-562q^{49}-14584q^{48}-23301q^{47}-11424q^{46}+9082q^{45}+20480q^{44}+18536q^{43}+1925q^{42}-12567q^{41}-23209q^{40}-13017q^{39}+6577q^{38}+18902q^{37}+18744q^{36}+4025q^{35}-9996q^{34}-21903q^{33}-13939q^{32}+3641q^{31}+16275q^{30}+18070q^{29}+6230q^{28}-6516q^{27}-19309q^{26}-14379q^{25}-17q^{24}+12256q^{23}+16248q^{22}+8380q^{21}-2039q^{20}-15052q^{19}-13741q^{18}-3837q^{17}+6926q^{16}+12722q^{15}+9457q^{14}+2570q^{13}-9316q^{12}-11221q^{11}-6390q^{10}+1428q^{9}+7634q^{8}+8378q^{7}+5645q^{6}-3440q^{5}-6936q^{4}-6427q^{3}-2386q^{2}+2408q+5274+5972q^{-1}+637q^{-2}-2421q^{-3}-4211q^{-4}-3375q^{-5}-1043q^{-6}+1772q^{-7}+4049q^{-8}+1940q^{-9}+469q^{-10}-1498q^{-11}-2226q^{-12}-1966q^{-13}-383q^{-14}+1704q^{-15}+1319q^{-16}+1203q^{-17}+109q^{-18}-710q^{-19}-1322q^{-20}-849q^{-21}+326q^{-22}+386q^{-23}+760q^{-24}+435q^{-25}+65q^{-26}-517q^{-27}-511q^{-28}-50q^{-29}-40q^{-30}+259q^{-31}+235q^{-32}+178q^{-33}-124q^{-34}-183q^{-35}-37q^{-36}-79q^{-37}+47q^{-38}+66q^{-39}+91q^{-40}-20q^{-41}-46q^{-42}-3q^{-43}-32q^{-44}+3q^{-45}+9q^{-46}+29q^{-47}-4q^{-48}-11q^{-49}+4q^{-50}-7q^{-51}+5q^{-54}-3q^{-56}+q^{-57}$
7	$q^{175}-3q^{174}+q^{173}+4q^{172}-3q^{171}-3q^{169}+5q^{168}+5q^{167}-15q^{166}+4q^{165}+10q^{164}-9q^{163}+q^{162}-9q^{161}+18q^{160}+30q^{159}-39q^{158}-4q^{157}+2q^{156}-38q^{155}+8q^{154}-18q^{153}+72q^{152}+128q^{151}-40q^{150}-32q^{149}-97q^{148}-196q^{147}-43q^{146}-29q^{145}+234q^{144}+475q^{143}+166q^{142}+29q^{141}-382q^{140}-777q^{139}-496q^{138}-299q^{137}+523q^{136}+1397q^{135}+1170q^{134}+775q^{133}-589q^{132}-2133q^{131}-2238q^{130}-1867q^{129}+209q^{128}+2976q^{127}+3962q^{126}+3859q^{125}+831q^{124}-3669q^{123}-6152q^{122}-6934q^{121}-3237q^{120}+3536q^{119}+8727q^{118}+11480q^{117}+7410q^{116}-2142q^{115}-11013q^{114}-17113q^{113}-13826q^{112}-1577q^{111}+12246q^{110}+23582q^{109}+22543q^{108}+8039q^{107}-11485q^{106}-29839q^{105}-33120q^{104}-17630q^{103}+7839q^{102}+34850q^{101}+44718q^{100}+30029q^{99}-850q^{98}-37563q^{97}-56110q^{96}-44371q^{95}-9338q^{94}+37217q^{93}+65925q^{92}+59393q^{91}+22152q^{90}-33574q^{89}-73308q^{88}-73745q^{87}-36224q^{86}+27132q^{85}+77469q^{84}+86020q^{83}+50450q^{82}-18532q^{81}-78644q^{80}-95665q^{79}-63379q^{78}+9165q^{77}+77112q^{76}+102096q^{75}+74261q^{74}+242q^{73}-73763q^{72}-105780q^{71}-82631q^{70}-8634q^{69}+69412q^{68}+107020q^{67}+88485q^{66}+15723q^{65}-64638q^{64}-106546q^{63}-92286q^{62}-21426q^{61}+59967q^{60}+104969q^{59}+94376q^{58}+25954q^{57}-55370q^{56}-102562q^{55}-95408q^{54}-29849q^{53}+50781q^{52}+99604q^{51}+95724q^{50}+33410q^{49}-45807q^{48}-95872q^{47}-95516q^{46}-37197q^{45}+40027q^{44}+91288q^{43}+94774q^{42}+41244q^{41}-33105q^{40}-85283q^{39}-93303q^{38}-45740q^{37}+24934q^{36}+77726q^{35}+90575q^{34}+50115q^{33}-15322q^{32}-68123q^{31}-86413q^{30}-54154q^{29}+5066q^{28}+56812q^{27}+79873q^{26}+56737q^{25}+5750q^{24}-43635q^{23}-71394q^{22}-57635q^{21}-15435q^{20}+29857q^{19}+60242q^{18}+55565q^{17}+23803q^{16}-15745q^{15}-47752q^{14}-51047q^{13}-29078q^{12}+3342q^{11}+33994q^{10}+43390q^{9}+31400q^{8}+7164q^{7}-20876q^{6}-34232q^{5}-30061q^{4}-13954q^{3}+9077q^{2}+23754q+26028+17594q^{-1}-74q^{-2}-14067q^{-3}-19912q^{-4}-17539q^{-5}-5931q^{-6}+5636q^{-7}+13249q^{-8}+15234q^{-9}+8625q^{-10}+251q^{-11}-6930q^{-12}-11303q^{-13}-8874q^{-14}-3797q^{-15}+2120q^{-16}+7236q^{-17}+7281q^{-18}+4985q^{-19}+1075q^{-20}-3583q^{-21}-5085q^{-22}-4758q^{-23}-2533q^{-24}+1132q^{-25}+2876q^{-26}+3557q^{-27}+2797q^{-28}+376q^{-29}-1181q^{-30}-2342q^{-31}-2380q^{-32}-861q^{-33}+201q^{-34}+1209q^{-35}+1622q^{-36}+918q^{-37}+334q^{-38}-511q^{-39}-1055q^{-40}-680q^{-41}-368q^{-42}+123q^{-43}+520q^{-44}+401q^{-45}+372q^{-46}+60q^{-47}-287q^{-48}-241q^{-49}-217q^{-50}-53q^{-51}+107q^{-52}+75q^{-53}+138q^{-54}+88q^{-55}-51q^{-56}-56q^{-57}-72q^{-58}-24q^{-59}+27q^{-60}-5q^{-61}+28q^{-62}+31q^{-63}-3q^{-64}-9q^{-65}-20q^{-66}-5q^{-67}+11q^{-68}-4q^{-69}+7q^{-71}-5q^{-74}+3q^{-76}-q^{-77}$

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[9, 26]]
Out[2]=	PD[X[1, 4, 2, 5], X[5, 12, 6, 13], X[3, 11, 4, 10], X[11, 3, 12, 2], X[13, 18, 14, 1], X[7, 15, 8, 14], X[17, 7, 18, 6], X[9, 17, 10, 16], X[15, 9, 16, 8]]
In[3]:=	GaussCode[Knot[9, 26]]
Out[3]=	GaussCode[-1, 4, -3, 1, -2, 7, -6, 9, -8, 3, -4, 2, -5, 6, -9, 8, -7, 5]
In[4]:=	DTCode[Knot[9, 26]]
Out[4]=	DTCode[4, 10, 12, 14, 16, 2, 18, 8, 6]
In[5]:=	br = BR[Knot[9, 26]]
Out[5]=	BR[4, {1, 1, 1, -2, 1, -2, 3, -2, 3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 9}
In[7]:=	BraidIndex[Knot[9, 26]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[9, 26]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[9, 26]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 3, 2, {4, 6}, 1}
In[10]:=	alex = Alexander[Knot[9, 26]][t]
Out[10]=	-3 5 11 2 3 -13 + t - -- + -- + 11 t - 5 t + t 2 t t
In[11]:=	Conway[Knot[9, 26]][z]
Out[11]=	4 6 1 + z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 26], Knot[11, NonAlternating, 25]}
In[13]:=	{KnotDet[Knot[9, 26]], KnotSignature[Knot[9, 26]]}
Out[13]=	{47, 2}
In[14]:=	Jones[Knot[9, 26]][q]
Out[14]=	-2 3 2 3 4 5 6 7 -4 - q + - + 7 q - 8 q + 8 q - 7 q + 5 q - 3 q + q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 26]}
In[16]:=	A2Invariant[Knot[9, 26]][q]
Out[16]=	-6 -4 2 4 6 8 14 16 18 22 1 - q + q + 3 q - q + 2 q - q - 2 q + q - q + q
In[17]:=	HOMFLYPT[Knot[9, 26]][a, z]
Out[17]=	2 2 2 4 4 6 -6 3 3 2 z 5 z 6 z 4 2 z 4 z z a - -- + -- - 2 z + -- - ---- + ---- - z - ---- + ---- + -- 4 2 6 4 2 4 2 2 a a a a a a a a
In[18]:=	Kauffman[Knot[9, 26]][a, z]
Out[18]=	2 2 2 2 -6 3 3 z z z z 2 z 2 z 11 z 13 z -a - -- - -- + -- + -- - -- - - + 5 z - -- + ---- + ----- + ----- - 4 2 7 5 3 a 8 6 4 2 a a a a a a a a a 3 3 3 3 4 4 4 4 z 2 z 7 z 3 z 3 4 z 5 z 14 z ---- - ---- + ---- + ---- - 2 a z - 8 z + -- - ---- - ----- - 7 5 3 a 8 6 4 a a a a a a 4 5 5 5 5 6 6 6 16 z 3 z z 11 z 6 z 5 6 4 z 6 z 5 z ----- + ---- - -- - ----- - ---- + a z + 3 z + ---- + ---- + ---- + 2 7 5 3 a 6 4 2 a a a a a a a 7 7 7 8 8 3 z 6 z 3 z z z ---- + ---- + ---- + -- + -- 5 3 a 4 2 a a a a
In[19]:=	{Vassiliev[2][Knot[9, 26]], Vassiliev[3][Knot[9, 26]]}
Out[19]=	{0, -1}
In[20]:=	Kh[Knot[9, 26]][q, t]
Out[20]=	3 1 2 1 2 2 q 3 5 5 q + 3 q + ----- + ----- + ---- + --- + --- + 4 q t + 4 q t + 5 3 3 2 2 q t t q t q t q t 5 2 7 2 7 3 9 3 9 4 11 4 11 5 4 q t + 4 q t + 3 q t + 4 q t + 2 q t + 3 q t + q t + 13 5 15 6 2 q t + q t
In[21]:=	ColouredJones[Knot[9, 26], 2][q]
Out[21]=	-7 3 9 11 5 26 2 3 4 -19 + q - -- + -- - -- - -- + -- - 19 q + 48 q - 23 q - 37 q + 6 4 3 2 q q q q q 5 6 7 8 9 10 11 12 64 q - 20 q - 49 q + 64 q - 12 q - 46 q + 48 q - 3 q - 13 14 15 16 17 18 19 20 31 q + 25 q + q - 14 q + 8 q + q - 3 q + q

See/edit the Rolfsen_Splice_Template. Back to the top.

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