K11n74

K11n73

K11n75

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n74 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X5,14,6,15 X2837 X9,20,10,21 X16,12,17,11 X13,6,14,7 X18,16,19,15 X12,18,13,17 X19,22,20,1 X21,10,22,11
Gauss code	1, -4, 2, -1, -3, 7, 4, -2, -5, 11, 6, -9, -7, 3, 8, -6, 9, -8, -10, 5, -11, 10
Dowker-Thistlethwaite code	4 8 -14 2 -20 16 -6 18 12 -22 -10

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 2 Bridge index 4 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n74/ThurstonBennequinNumber Hyperbolic Volume 10.4045 A-Polynomial See Data:K11n74/A-polynomial

[edit Notes for K11n74's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n74's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{2}-2t+3-2t^{-1}+t^{-2}$
Conway polynomial	$z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\left\{t^{2}-t+1\right\}$
Determinant and Signature	{ 9, 0 }
Jones polynomial	$q^{6}-2q^{5}+2q^{4}-3q^{3}+2q^{2}-q+2+q^{-1}-q^{-2}+q^{-3}-q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-2}+z^{6}-a^{2}z^{4}-6z^{4}a^{-2}+z^{4}a^{-4}+7z^{4}-4a^{2}z^{2}-12z^{2}a^{-2}+3z^{2}a^{-4}+15z^{2}-4a^{2}-8a^{-2}+2a^{-4}+11$
Kauffman polynomial (db, data sources)	$a^{2}z^{8}+z^{8}a^{-2}+z^{8}a^{-4}+z^{8}+a^{3}z^{7}+2az^{7}+2z^{7}a^{-1}+3z^{7}a^{-3}+2z^{7}a^{-5}-6a^{2}z^{6}-6z^{6}a^{-2}-3z^{6}a^{-4}+z^{6}a^{-6}-8z^{6}-6a^{3}z^{5}-14az^{5}-15z^{5}a^{-1}-16z^{5}a^{-3}-9z^{5}a^{-5}+10a^{2}z^{4}+11z^{4}a^{-2}-z^{4}a^{-4}-4z^{4}a^{-6}+18z^{4}+10a^{3}z^{3}+25az^{3}+29z^{3}a^{-1}+24z^{3}a^{-3}+10z^{3}a^{-5}-8a^{2}z^{2}-13z^{2}a^{-2}+z^{2}a^{-4}+3z^{2}a^{-6}-19z^{2}-5a^{3}z-13az-17za^{-1}-13za^{-3}-4za^{-5}+4a^{2}+8a^{-2}+2a^{-4}+11$
The A2 invariant	Data:K11n74/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n74/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n74 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["K11n74"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{2}-2t+3-2t^{-1}+t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

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

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 9, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{6}a^{-2}+z^{6}-a^{2}z^{4}-6z^{4}a^{-2}+z^{4}a^{-4}+7z^{4}-4a^{2}z^{2}-12z^{2}a^{-2}+3z^{2}a^{-4}+15z^{2}-4a^{2}-8a^{-2}+2a^{-4}+11$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{2}z^{8}+z^{8}a^{-2}+z^{8}a^{-4}+z^{8}+a^{3}z^{7}+2az^{7}+2z^{7}a^{-1}+3z^{7}a^{-3}+2z^{7}a^{-5}-6a^{2}z^{6}-6z^{6}a^{-2}-3z^{6}a^{-4}+z^{6}a^{-6}-8z^{6}-6a^{3}z^{5}-14az^{5}-15z^{5}a^{-1}-16z^{5}a^{-3}-9z^{5}a^{-5}+10a^{2}z^{4}+11z^{4}a^{-2}-z^{4}a^{-4}-4z^{4}a^{-6}+18z^{4}+10a^{3}z^{3}+25az^{3}+29z^{3}a^{-1}+24z^{3}a^{-3}+10z^{3}a^{-5}-8a^{2}z^{2}-13z^{2}a^{-2}+z^{2}a^{-4}+3z^{2}a^{-6}-19z^{2}-5a^{3}z-13az-17za^{-1}-13za^{-3}-4za^{-5}+4a^{2}+8a^{-2}+2a^{-4}+11$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_20, 10_140, K11n73,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n73,}

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["K11n74"];

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]=

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

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]=

{8_20, 10_140, K11n73,}

In[6]:=

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

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

Out[6]=

{K11n73,}

Vassiliev invariants

V2 and V3:

(2, -2)

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 $8$ $-16$ $32$ ${\frac {76}{3}}$ ${\frac {20}{3}}$ $-128$ $-{\frac {544}{3}}$ ${\frac {32}{3}}$ $-48$ ${\frac {256}{3}}$ $128$ ${\frac {608}{3}}$ ${\frac {160}{3}}$ ${\frac {6391}{15}}$ $-{\frac {308}{5}}$ ${\frac {12604}{45}}$ ${\frac {233}{9}}$ ${\frac {631}{15}}$

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 K11n74. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11n73 K11n75