K11a80

K11a79

K11a81

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a80 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X12,6,13,5 X14,7,15,8 X22,10,1,9 X2,11,3,12 X18,14,19,13 X20,15,21,16 X6,17,7,18 X8,20,9,19 X16,21,17,22
Gauss code	1, -6, 2, -1, 3, -9, 4, -10, 5, -2, 6, -3, 7, -4, 8, -11, 9, -7, 10, -8, 11, -5
Dowker-Thistlethwaite code	4 10 12 14 22 2 18 20 6 8 16

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 1 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a80/ThurstonBennequinNumber Hyperbolic Volume 16.5161 A-Polynomial See Data:K11a80/A-polynomial

[edit Notes for K11a80's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $4$ Rasmussen s-Invariant 0

[edit Notes for K11a80's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-6t^{3}+16t^{2}-28t+35-28t^{-1}+16t^{-2}-6t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+2z^{6}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 137, 0 }
Jones polynomial	$-q^{5}+4q^{4}-8q^{3}+14q^{2}-19q+22-22q^{-1}+19q^{-2}-14q^{-3}+9q^{-4}-4q^{-5}+q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}-2a^{2}z^{6}-z^{6}a^{-2}+5z^{6}+a^{4}z^{4}-7a^{2}z^{4}-3z^{4}a^{-2}+9z^{4}+2a^{4}z^{2}-7a^{2}z^{2}-2z^{2}a^{-2}+5z^{2}+a^{4}-a^{2}+a^{-2}$
Kauffman polynomial (db, data sources)	$2a^{2}z^{10}+2z^{10}+6a^{3}z^{9}+12az^{9}+6z^{9}a^{-1}+7a^{4}z^{8}+12a^{2}z^{8}+8z^{8}a^{-2}+13z^{8}+4a^{5}z^{7}-7a^{3}z^{7}-19az^{7}-z^{7}a^{-1}+7z^{7}a^{-3}+a^{6}z^{6}-16a^{4}z^{6}-38a^{2}z^{6}-9z^{6}a^{-2}+4z^{6}a^{-4}-34z^{6}-9a^{5}z^{5}-7a^{3}z^{5}+az^{5}-12z^{5}a^{-1}-10z^{5}a^{-3}+z^{5}a^{-5}-2a^{6}z^{4}+10a^{4}z^{4}+33a^{2}z^{4}-6z^{4}a^{-4}+27z^{4}+5a^{5}z^{3}+8a^{3}z^{3}+11az^{3}+13z^{3}a^{-1}+4z^{3}a^{-3}-z^{3}a^{-5}+a^{6}z^{2}-4a^{4}z^{2}-11a^{2}z^{2}+3z^{2}a^{-2}+2z^{2}a^{-4}-5z^{2}-a^{5}z-2a^{3}z-4az-4za^{-1}-za^{-3}+a^{4}+a^{2}-a^{-2}$
The A2 invariant	$q^{18}-q^{16}+2q^{12}-3q^{10}+4q^{8}-q^{6}-q^{4}+2q^{2}-5+4q^{-2}-3q^{-4}+2q^{-6}+3q^{-8}-2q^{-10}+2q^{-12}-q^{-14}$
The G2 invariant	$q^{94}-3q^{92}+8q^{90}-16q^{88}+22q^{86}-24q^{84}+12q^{82}+20q^{80}-66q^{78}+122q^{76}-163q^{74}+153q^{72}-74q^{70}-81q^{68}+278q^{66}-435q^{64}+489q^{62}-371q^{60}+80q^{58}+297q^{56}-636q^{54}+791q^{52}-674q^{50}+309q^{48}+174q^{46}-588q^{44}+763q^{42}-624q^{40}+243q^{38}+218q^{36}-548q^{34}+593q^{32}-336q^{30}-117q^{28}+568q^{26}-804q^{24}+717q^{22}-313q^{20}-267q^{18}+803q^{16}-1097q^{14}+1025q^{12}-605q^{10}-21q^{8}+624q^{6}-994q^{4}+998q^{2}-656+120q^{-2}+386q^{-4}-666q^{-6}+614q^{-8}-284q^{-10}-153q^{-12}+499q^{-14}-586q^{-16}+384q^{-18}+11q^{-20}-425q^{-22}+690q^{-24}-694q^{-26}+460q^{-28}-82q^{-30}-296q^{-32}+542q^{-34}-600q^{-36}+486q^{-38}-256q^{-40}+10q^{-42}+184q^{-44}-289q^{-46}+294q^{-48}-230q^{-50}+132q^{-52}-31q^{-54}-46q^{-56}+86q^{-58}-95q^{-60}+77q^{-62}-46q^{-64}+20q^{-66}+3q^{-68}-14q^{-70}+15q^{-72}-13q^{-74}+7q^{-76}-3q^{-78}+q^{-80}$

Further Quantum Invariants

K11a80 Quantum Invariants.

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["K11a80"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-6t^{3}+16t^{2}-28t+35-28t^{-1}+16t^{-2}-6t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{8}+2z^{6}-2z^{2}+1$

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]=

{ 137, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{5}+4q^{4}-8q^{3}+14q^{2}-19q+22-22q^{-1}+19q^{-2}-14q^{-3}+9q^{-4}-4q^{-5}+q^{-6}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2a^{2}z^{10}+2z^{10}+6a^{3}z^{9}+12az^{9}+6z^{9}a^{-1}+7a^{4}z^{8}+12a^{2}z^{8}+8z^{8}a^{-2}+13z^{8}+4a^{5}z^{7}-7a^{3}z^{7}-19az^{7}-z^{7}a^{-1}+7z^{7}a^{-3}+a^{6}z^{6}-16a^{4}z^{6}-38a^{2}z^{6}-9z^{6}a^{-2}+4z^{6}a^{-4}-34z^{6}-9a^{5}z^{5}-7a^{3}z^{5}+az^{5}-12z^{5}a^{-1}-10z^{5}a^{-3}+z^{5}a^{-5}-2a^{6}z^{4}+10a^{4}z^{4}+33a^{2}z^{4}-6z^{4}a^{-4}+27z^{4}+5a^{5}z^{3}+8a^{3}z^{3}+11az^{3}+13z^{3}a^{-1}+4z^{3}a^{-3}-z^{3}a^{-5}+a^{6}z^{2}-4a^{4}z^{2}-11a^{2}z^{2}+3z^{2}a^{-2}+2z^{2}a^{-4}-5z^{2}-a^{5}z-2a^{3}z-4az-4za^{-1}-za^{-3}+a^{4}+a^{2}-a^{-2}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a270,}

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["K11a80"];

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]=

{ $t^{4}-6t^{3}+16t^{2}-28t+35-28t^{-1}+16t^{-2}-6t^{-3}+t^{-4}$, $-q^{5}+4q^{4}-8q^{3}+14q^{2}-19q+22-22q^{-1}+19q^{-2}-14q^{-3}+9q^{-4}-4q^{-5}+q^{-6}$ }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{K11a270,}

Vassiliev invariants

V2 and V3:

(-2, 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 $-8$ $8$ $32$ ${\frac {116}{3}}$ ${\frac {52}{3}}$ $-64$ $-{\frac {400}{3}}$ $-{\frac {160}{3}}$ $-24$ $-{\frac {256}{3}}$ $32$ $-{\frac {928}{3}}$ $-{\frac {416}{3}}$ $-{\frac {2311}{15}}$ $-{\frac {796}{15}}$ $-{\frac {2884}{45}}$ ${\frac {199}{9}}$ $-{\frac {151}{15}}$

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=$0 is the signature of K11a80. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     3   3 7                   5 1   -4 5                 9 3     6 3               10 5       -5 1             12 9         3 -1           11 11           0 -3         8 11             -3 -5       6 11               5 -7     3 8                 -5 -9   1 6                   5 -11   3                     -3 -13 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{12}$ $r=1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

Computer Talk

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

Modifying This Page

Read me first: Modifying Knot Pages. See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate). See/edit the Hoste-Thistlethwaite_Splice_Base (expert). Back to the top.

K11a79 K11a81