K11n77

K11n76

K11n78

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n77 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X5,15,6,14 X2837 X20,10,21,9 X11,17,12,16 X13,19,14,18 X15,7,16,6 X17,13,18,12 X22,20,1,19 X10,22,11,21
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 4 3-genus 4 Bridge index 4 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n77/ThurstonBennequinNumber Hyperbolic Volume 11.9604 A-Polynomial See Data:K11n77/A-polynomial

[edit Notes for K11n77's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $4$ Rasmussen s-Invariant 8

[edit Notes for K11n77's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-t^{3}-2t^{2}+8t-11+8t^{-1}-2t^{-2}-t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+7z^{6}+12z^{4}+7z^{2}+1$
2nd Alexander ideal (db, data sources)	$\left\{t^{2}-t+1\right\}$
Determinant and Signature	{ 27, 6 }
Jones polynomial	$-q^{14}+3q^{13}-4q^{12}+5q^{11}-6q^{10}+4q^{9}-4q^{8}+2q^{7}+q^{6}+q^{4}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}a^{-8}+8z^{6}a^{-8}-z^{6}a^{-10}+21z^{4}a^{-8}-9z^{4}a^{-10}+23z^{2}a^{-8}-20z^{2}a^{-10}+4z^{2}a^{-12}+9a^{-8}-13a^{-10}+6a^{-12}-a^{-14}$
Kauffman polynomial (db, data sources)	$z^{8}a^{-8}+z^{8}a^{-10}+z^{8}a^{-12}+z^{8}a^{-14}+z^{7}a^{-9}+2z^{7}a^{-11}+4z^{7}a^{-13}+3z^{7}a^{-15}-8z^{6}a^{-8}-10z^{6}a^{-10}-4z^{6}a^{-12}+z^{6}a^{-14}+3z^{6}a^{-16}-9z^{5}a^{-9}-15z^{5}a^{-11}-15z^{5}a^{-13}-8z^{5}a^{-15}+z^{5}a^{-17}+21z^{4}a^{-8}+29z^{4}a^{-10}+8z^{4}a^{-12}-8z^{4}a^{-14}-8z^{4}a^{-16}+20z^{3}a^{-9}+36z^{3}a^{-11}+23z^{3}a^{-13}+5z^{3}a^{-15}-2z^{3}a^{-17}-23z^{2}a^{-8}-31z^{2}a^{-10}-6z^{2}a^{-12}+5z^{2}a^{-14}+3z^{2}a^{-16}-13za^{-9}-22za^{-11}-12za^{-13}-3za^{-15}+9a^{-8}+13a^{-10}+6a^{-12}+a^{-14}$
The A2 invariant	$q^{-14}+q^{-16}+2q^{-18}+4q^{-20}+2q^{-22}+q^{-24}-5q^{-28}-3q^{-30}-4q^{-32}+2q^{-36}+q^{-38}+3q^{-40}-q^{-42}-q^{-44}$
The G2 invariant	Data:K11n77/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n77 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["K11n77"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-t^{3}-2t^{2}+8t-11+8t^{-1}-2t^{-2}-t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{8}+7z^{6}+12z^{4}+7z^{2}+1$

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]=

{ 27, 6 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{14}+3q^{13}-4q^{12}+5q^{11}-6q^{10}+4q^{9}-4q^{8}+2q^{7}+q^{6}+q^{4}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}a^{-8}+8z^{6}a^{-8}-z^{6}a^{-10}+21z^{4}a^{-8}-9z^{4}a^{-10}+23z^{2}a^{-8}-20z^{2}a^{-10}+4z^{2}a^{-12}+9a^{-8}-13a^{-10}+6a^{-12}-a^{-14}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{8}a^{-8}+z^{8}a^{-10}+z^{8}a^{-12}+z^{8}a^{-14}+z^{7}a^{-9}+2z^{7}a^{-11}+4z^{7}a^{-13}+3z^{7}a^{-15}-8z^{6}a^{-8}-10z^{6}a^{-10}-4z^{6}a^{-12}+z^{6}a^{-14}+3z^{6}a^{-16}-9z^{5}a^{-9}-15z^{5}a^{-11}-15z^{5}a^{-13}-8z^{5}a^{-15}+z^{5}a^{-17}+21z^{4}a^{-8}+29z^{4}a^{-10}+8z^{4}a^{-12}-8z^{4}a^{-14}-8z^{4}a^{-16}+20z^{3}a^{-9}+36z^{3}a^{-11}+23z^{3}a^{-13}+5z^{3}a^{-15}-2z^{3}a^{-17}-23z^{2}a^{-8}-31z^{2}a^{-10}-6z^{2}a^{-12}+5z^{2}a^{-14}+3z^{2}a^{-16}-13za^{-9}-22za^{-11}-12za^{-13}-3za^{-15}+9a^{-8}+13a^{-10}+6a^{-12}+a^{-14}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

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["K11n77"];

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]=

{ $t^{4}-t^{3}-2t^{2}+8t-11+8t^{-1}-2t^{-2}-t^{-3}+t^{-4}$, $-q^{14}+3q^{13}-4q^{12}+5q^{11}-6q^{10}+4q^{9}-4q^{8}+2q^{7}+q^{6}+q^{4}$ }

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]=

{}

Vassiliev invariants

V2 and V3:

(7, 16)

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 $28$ $128$ $392$ ${\frac {2354}{3}}$ ${\frac {286}{3}}$ $3584$ ${\frac {16256}{3}}$ ${\frac {2624}{3}}$ $544$ ${\frac {10976}{3}}$ $8192$ ${\frac {65912}{3}}$ ${\frac {8008}{3}}$ ${\frac {1164937}{30}}$ ${\frac {12922}{5}}$ ${\frac {531554}{45}}$ ${\frac {4439}{18}}$ ${\frac {41257}{30}}$

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=$6 is the signature of K11n77. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ 0 1 2 3 4 5 6 7 8 9 10 11 χ 29                       1 -1 27                     2   2 25                   2 1   -1 23                 3 2     1 21               3 2       -1 19           1 2 3         -2 17           3 3           0 15       1 1 2             -2 13         3               3 11     1                   1 9 1                       1 7 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=5$ $i=7$ $i=9$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=1$ $r=2$ ${\mathbb {Z} }$ $r=3$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=4$ ${\mathbb {Z} }^{3}$ ${\mathbb {Z} }$ $r=5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ ${\mathbb {Z} }$ $r=6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=7$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=8$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=9$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=10$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=11$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11n76 K11n78