K11n76

K11n75

K11n77

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n76 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X14,5,15,6 X2837 X9,20,10,21 X16,11,17,12 X18,13,19,14 X6,15,7,16 X12,17,13,18 X19,22,20,1 X21,10,22,11
Gauss code	1, -4, 2, -1, 3, -8, 4, -2, -5, 11, 6, -9, 7, -3, 8, -6, 9, -7, -10, 5, -11, 10
Dowker-Thistlethwaite code	4 8 14 2 -20 16 18 6 12 -22 -10

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 3 3-genus 4 Bridge index 4 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n76/ThurstonBennequinNumber Hyperbolic Volume 12.3125 A-Polynomial See Data:K11n76/A-polynomial

[edit Notes for K11n76's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $2$ Rasmussen s-Invariant 4

[edit Notes for K11n76's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-3t^{3}+6t^{2}-8t+9-8t^{-1}+6t^{-2}-3t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+5z^{6}+8z^{4}+5z^{2}+1$
2nd Alexander ideal (db, data sources)	$\left\{t^{2}-t+1\right\}$
Determinant and Signature	{ 45, -4 }
Jones polynomial	$-q+2-4q^{-1}+7q^{-2}-6q^{-3}+8q^{-4}-7q^{-5}+5q^{-6}-4q^{-7}+q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$z^{2}a^{8}+2a^{8}-z^{6}a^{6}-6z^{4}a^{6}-13z^{2}a^{6}-10a^{6}+z^{8}a^{4}+7z^{6}a^{4}+19z^{4}a^{4}+25z^{2}a^{4}+13a^{4}-z^{6}a^{2}-5z^{4}a^{2}-8z^{2}a^{2}-4a^{2}$
Kauffman polynomial (db, data sources)	$z^{2}a^{10}+4z^{3}a^{9}-2za^{9}+2z^{6}a^{8}-2z^{4}a^{8}+z^{2}a^{8}+2a^{8}+5z^{7}a^{7}-18z^{5}a^{7}+26z^{3}a^{7}-14za^{7}+4z^{8}a^{6}-13z^{6}a^{6}+16z^{4}a^{6}-17z^{2}a^{6}+10a^{6}+z^{9}a^{5}+5z^{7}a^{5}-31z^{5}a^{5}+42z^{3}a^{5}-21za^{5}+6z^{8}a^{4}-24z^{6}a^{4}+31z^{4}a^{4}-26z^{2}a^{4}+13a^{4}+z^{9}a^{3}+z^{7}a^{3}-18z^{5}a^{3}+28z^{3}a^{3}-13za^{3}+2z^{8}a^{2}-9z^{6}a^{2}+13z^{4}a^{2}-9z^{2}a^{2}+4a^{2}+z^{7}a-5z^{5}a+8z^{3}a-4za$
The A2 invariant	$q^{28}+q^{24}-3q^{22}-3q^{20}-3q^{18}-2q^{16}+4q^{14}+2q^{12}+6q^{10}+2q^{8}+q^{6}-2q^{2}-q^{-2}$
The G2 invariant	Data:K11n76/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n76 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["K11n76"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-3t^{3}+6t^{2}-8t+9-8t^{-1}+6t^{-2}-3t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{8}+5z^{6}+8z^{4}+5z^{2}+1$

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]=

$\left\{t^{2}-t+1\right\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 45, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{2}a^{8}+2a^{8}-z^{6}a^{6}-6z^{4}a^{6}-13z^{2}a^{6}-10a^{6}+z^{8}a^{4}+7z^{6}a^{4}+19z^{4}a^{4}+25z^{2}a^{4}+13a^{4}-z^{6}a^{2}-5z^{4}a^{2}-8z^{2}a^{2}-4a^{2}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{2}a^{10}+4z^{3}a^{9}-2za^{9}+2z^{6}a^{8}-2z^{4}a^{8}+z^{2}a^{8}+2a^{8}+5z^{7}a^{7}-18z^{5}a^{7}+26z^{3}a^{7}-14za^{7}+4z^{8}a^{6}-13z^{6}a^{6}+16z^{4}a^{6}-17z^{2}a^{6}+10a^{6}+z^{9}a^{5}+5z^{7}a^{5}-31z^{5}a^{5}+42z^{3}a^{5}-21za^{5}+6z^{8}a^{4}-24z^{6}a^{4}+31z^{4}a^{4}-26z^{2}a^{4}+13a^{4}+z^{9}a^{3}+z^{7}a^{3}-18z^{5}a^{3}+28z^{3}a^{3}-13za^{3}+2z^{8}a^{2}-9z^{6}a^{2}+13z^{4}a^{2}-9z^{2}a^{2}+4a^{2}+z^{7}a-5z^{5}a+8z^{3}a-4za$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_62, K11n78,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n78,}

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["K11n76"];

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]=

{ $t^{4}-3t^{3}+6t^{2}-8t+9-8t^{-1}+6t^{-2}-3t^{-3}+t^{-4}$, $-q+2-4q^{-1}+7q^{-2}-6q^{-3}+8q^{-4}-7q^{-5}+5q^{-6}-4q^{-7}+q^{-8}$ }

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]=

{10_62, K11n78,}

In[6]:=

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

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

Out[6]=

{K11n78,}

Vassiliev invariants

V2 and V3:

(5, -7)

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 $20$ $-56$ $200$ ${\frac {886}{3}}$ ${\frac {98}{3}}$ $-1120$ $-{\frac {4496}{3}}$ $-{\frac {704}{3}}$ $-152$ ${\frac {4000}{3}}$ $1568$ ${\frac {17720}{3}}$ ${\frac {1960}{3}}$ ${\frac {50767}{6}}$ ${\frac {2090}{3}}$ ${\frac {21446}{9}}$ ${\frac {853}{18}}$ ${\frac {1423}{6}}$

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 K11n76. 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 χ 3                   1 -1 1                 1   1 -1               3 1   -2 -3             4 1     3 -5           3 4       1 -7         5 3         2 -9       2 3           1 -11     3 5             -2 -13   1 2               1 -15   3                 -3 -17 1                   1

Integral Khovanov Homology (db, data source)

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

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.

K11n75 K11n77