K11a76

K11a75

K11a77

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a76 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{1,2\}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a76/ThurstonBennequinNumber Hyperbolic Volume 16.9805 A-Polynomial See Data:K11a76/A-polynomial

[edit Notes for K11a76'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 K11a76's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-6t^{3}+17t^{2}-30t+37-30t^{-1}+17t^{-2}-6t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+2z^{6}+z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 145, 0 }
Jones polynomial	$-q^{5}+4q^{4}-9q^{3}+15q^{2}-20q+24-23q^{-1}+20q^{-2}-15q^{-3}+9q^{-4}-4q^{-5}+q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}-2a^{2}z^{6}-z^{6}a^{-2}+5z^{6}+a^{4}z^{4}-7a^{2}z^{4}-3z^{4}a^{-2}+10z^{4}+2a^{4}z^{2}-8a^{2}z^{2}-3z^{2}a^{-2}+9z^{2}+a^{4}-3a^{2}-a^{-2}+4$
Kauffman polynomial (db, data sources)	$2a^{2}z^{10}+2z^{10}+6a^{3}z^{9}+13az^{9}+7z^{9}a^{-1}+7a^{4}z^{8}+14a^{2}z^{8}+10z^{8}a^{-2}+17z^{8}+4a^{5}z^{7}-5a^{3}z^{7}-19az^{7}-2z^{7}a^{-1}+8z^{7}a^{-3}+a^{6}z^{6}-15a^{4}z^{6}-41a^{2}z^{6}-16z^{6}a^{-2}+4z^{6}a^{-4}-45z^{6}-9a^{5}z^{5}-10a^{3}z^{5}+az^{5}-11z^{5}a^{-1}-12z^{5}a^{-3}+z^{5}a^{-5}-2a^{6}z^{4}+9a^{4}z^{4}+35a^{2}z^{4}+12z^{4}a^{-2}-5z^{4}a^{-4}+41z^{4}+6a^{5}z^{3}+9a^{3}z^{3}+6az^{3}+10z^{3}a^{-1}+6z^{3}a^{-3}-z^{3}a^{-5}+a^{6}z^{2}-3a^{4}z^{2}-15a^{2}z^{2}-5z^{2}a^{-2}+z^{2}a^{-4}-17z^{2}-a^{5}z-2a^{3}z-2az-2za^{-1}-za^{-3}+a^{4}+3a^{2}+a^{-2}+4$
The A2 invariant	$q^{18}-q^{16}+2q^{12}-4q^{10}+3q^{8}-2q^{6}-q^{4}+4q^{2}-3+6q^{-2}-3q^{-4}+q^{-6}+2q^{-8}-3q^{-10}+2q^{-12}-q^{-14}$
The G2 invariant	$q^{94}-3q^{92}+8q^{90}-16q^{88}+22q^{86}-25q^{84}+14q^{82}+18q^{80}-67q^{78}+130q^{76}-176q^{74}+169q^{72}-88q^{70}-77q^{68}+294q^{66}-480q^{64}+556q^{62}-446q^{60}+129q^{58}+307q^{56}-717q^{54}+935q^{52}-836q^{50}+440q^{48}+134q^{46}-670q^{44}+941q^{42}-842q^{40}+406q^{38}+170q^{36}-633q^{34}+764q^{32}-509q^{30}-20q^{28}+597q^{26}-952q^{24}+914q^{22}-475q^{20}-231q^{18}+920q^{16}-1333q^{14}+1302q^{12}-812q^{10}+53q^{8}+721q^{6}-1226q^{4}+1288q^{2}-905+241q^{-2}+429q^{-4}-839q^{-6}+843q^{-8}-458q^{-10}-93q^{-12}+573q^{-14}-751q^{-16}+553q^{-18}-85q^{-20}-462q^{-22}+844q^{-24}-903q^{-26}+640q^{-28}-155q^{-30}-349q^{-32}+694q^{-34}-793q^{-36}+646q^{-38}-344q^{-40}+3q^{-42}+259q^{-44}-394q^{-46}+389q^{-48}-285q^{-50}+149q^{-52}-15q^{-54}-76q^{-56}+114q^{-58}-115q^{-60}+84q^{-62}-46q^{-64}+16q^{-66}+7q^{-68}-16q^{-70}+16q^{-72}-13q^{-74}+7q^{-76}-3q^{-78}+q^{-80}$

Further Quantum Invariants

K11a76 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["K11a76"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-6t^{3}+17t^{2}-30t+37-30t^{-1}+17t^{-2}-6t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{8}+2z^{6}+z^{4}+1$

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]=

{ 145, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{5}+4q^{4}-9q^{3}+15q^{2}-20q+24-23q^{-1}+20q^{-2}-15q^{-3}+9q^{-4}-4q^{-5}+q^{-6}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2a^{2}z^{10}+2z^{10}+6a^{3}z^{9}+13az^{9}+7z^{9}a^{-1}+7a^{4}z^{8}+14a^{2}z^{8}+10z^{8}a^{-2}+17z^{8}+4a^{5}z^{7}-5a^{3}z^{7}-19az^{7}-2z^{7}a^{-1}+8z^{7}a^{-3}+a^{6}z^{6}-15a^{4}z^{6}-41a^{2}z^{6}-16z^{6}a^{-2}+4z^{6}a^{-4}-45z^{6}-9a^{5}z^{5}-10a^{3}z^{5}+az^{5}-11z^{5}a^{-1}-12z^{5}a^{-3}+z^{5}a^{-5}-2a^{6}z^{4}+9a^{4}z^{4}+35a^{2}z^{4}+12z^{4}a^{-2}-5z^{4}a^{-4}+41z^{4}+6a^{5}z^{3}+9a^{3}z^{3}+6az^{3}+10z^{3}a^{-1}+6z^{3}a^{-3}-z^{3}a^{-5}+a^{6}z^{2}-3a^{4}z^{2}-15a^{2}z^{2}-5z^{2}a^{-2}+z^{2}a^{-4}-17z^{2}-a^{5}z-2a^{3}z-2az-2za^{-1}-za^{-3}+a^{4}+3a^{2}+a^{-2}+4$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a160, K11a289,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a160, K11a289,}

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["K11a76"];

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]=

{ $t^{4}-6t^{3}+17t^{2}-30t+37-30t^{-1}+17t^{-2}-6t^{-3}+t^{-4}$, $-q^{5}+4q^{4}-9q^{3}+15q^{2}-20q+24-23q^{-1}+20q^{-2}-15q^{-3}+9q^{-4}-4q^{-5}+q^{-6}$ }

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]=

{K11a160, K11a289,}

In[6]:=

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

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

Out[6]=

{K11a160, K11a289,}

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 $0$ $8$ $0$ $-16$ $-8$ $0$ $-{\frac {16}{3}}$ $-{\frac {160}{3}}$ $40$ $0$ $32$ $0$ $0$ $72$ $-24$ ${\frac {344}{3}}$ $-{\frac {136}{3}}$ $24$

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 K11a76. 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                   6 1   -5 5                 9 3     6 3               11 6       -5 1             13 9         4 -1           11 12           1 -3         9 12             -3 -5       6 11               5 -7     3 9                 -6 -9   1 6                   5 -11   3                     -3 -13 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-1$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=0$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{13}$ $r=1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a75 K11a77