K11n78

K11n77

K11n79

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n78 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X5,15,6,14 X2837 X9,20,10,21 X11,17,12,16 X13,19,14,18 X15,7,16,6 X17,13,18,12 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 $\{2,3\}$ 3-genus 4 Bridge index 4 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n78/ThurstonBennequinNumber Hyperbolic Volume 12.3125 A-Polynomial See Data:K11n78/A-polynomial

[edit Notes for K11n78'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 K11n78'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	$q^{8}-4q^{7}+5q^{6}-7q^{5}+8q^{4}-6q^{3}+7q^{2}-4q+2-q^{-1}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}a^{-4}-z^{6}a^{-2}+7z^{6}a^{-4}-z^{6}a^{-6}-5z^{4}a^{-2}+19z^{4}a^{-4}-6z^{4}a^{-6}-8z^{2}a^{-2}+25z^{2}a^{-4}-13z^{2}a^{-6}+z^{2}a^{-8}-4a^{-2}+13a^{-4}-10a^{-6}+2a^{-8}$
Kauffman polynomial (db, data sources)	$z^{9}a^{-3}+z^{9}a^{-5}+2z^{8}a^{-2}+6z^{8}a^{-4}+4z^{8}a^{-6}+z^{7}a^{-1}+z^{7}a^{-3}+5z^{7}a^{-5}+5z^{7}a^{-7}-9z^{6}a^{-2}-24z^{6}a^{-4}-13z^{6}a^{-6}+2z^{6}a^{-8}-5z^{5}a^{-1}-18z^{5}a^{-3}-31z^{5}a^{-5}-18z^{5}a^{-7}+13z^{4}a^{-2}+31z^{4}a^{-4}+16z^{4}a^{-6}-2z^{4}a^{-8}+8z^{3}a^{-1}+28z^{3}a^{-3}+42z^{3}a^{-5}+26z^{3}a^{-7}+4z^{3}a^{-9}-9z^{2}a^{-2}-26z^{2}a^{-4}-17z^{2}a^{-6}+z^{2}a^{-8}+z^{2}a^{-10}-4za^{-1}-13za^{-3}-21za^{-5}-14za^{-7}-2za^{-9}+4a^{-2}+13a^{-4}+10a^{-6}+2a^{-8}$
The A2 invariant	$-q^{2}-2q^{-2}+q^{-6}+2q^{-8}+6q^{-10}+2q^{-12}+4q^{-14}-2q^{-16}-3q^{-18}-3q^{-20}-3q^{-22}+q^{-24}+q^{-28}$
The G2 invariant	Data:K11n78/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n78 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["K11n78"];

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]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{9}a^{-3}+z^{9}a^{-5}+2z^{8}a^{-2}+6z^{8}a^{-4}+4z^{8}a^{-6}+z^{7}a^{-1}+z^{7}a^{-3}+5z^{7}a^{-5}+5z^{7}a^{-7}-9z^{6}a^{-2}-24z^{6}a^{-4}-13z^{6}a^{-6}+2z^{6}a^{-8}-5z^{5}a^{-1}-18z^{5}a^{-3}-31z^{5}a^{-5}-18z^{5}a^{-7}+13z^{4}a^{-2}+31z^{4}a^{-4}+16z^{4}a^{-6}-2z^{4}a^{-8}+8z^{3}a^{-1}+28z^{3}a^{-3}+42z^{3}a^{-5}+26z^{3}a^{-7}+4z^{3}a^{-9}-9z^{2}a^{-2}-26z^{2}a^{-4}-17z^{2}a^{-6}+z^{2}a^{-8}+z^{2}a^{-10}-4za^{-1}-13za^{-3}-21za^{-5}-14za^{-7}-2za^{-9}+4a^{-2}+13a^{-4}+10a^{-6}+2a^{-8}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_62, K11n76,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n76,}

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["K11n78"];

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]=

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

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, K11n76,}

In[6]:=

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

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

Out[6]=

{K11n76,}

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 K11n78. 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 17                   1 1 15                 3   -3 13               2 1   1 11             5 3     -2 9           3 2       1 7         3 5         2 5       4 3           1 3     1 4             3 1   1 3               -2 -1   1                 1 -3 1                   -1

Integral Khovanov Homology (db, data source)

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

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.

K11n77 K11n79