K11n84

K11n83

K11n85

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n84 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8394 X5,18,6,19 X7,12,8,13 X2,9,3,10 X11,17,12,16 X13,20,14,21 X15,6,16,7 X17,11,18,10 X19,22,20,1 X21,14,22,15
Gauss code	1, -5, 2, -1, -3, 8, -4, -2, 5, 9, -6, 4, -7, 11, -8, 6, -9, 3, -10, 7, -11, 10
Dowker-Thistlethwaite code	4 8 -18 -12 2 -16 -20 -6 -10 -22 -14

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n84/ThurstonBennequinNumber Hyperbolic Volume 11.4316 A-Polynomial See Data:K11n84/A-polynomial

[edit Notes for K11n84's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n84's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{2}+9t-13+9t^{-1}-2t^{-2}$
Conway polynomial	$-2z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 35, -2 }
Jones polynomial	$2q^{-1}-3q^{-2}+5q^{-3}-6q^{-4}+6q^{-5}-5q^{-6}+4q^{-7}-3q^{-8}+q^{-9}$
HOMFLY-PT polynomial (db, data sources)	$z^{2}a^{8}-z^{4}a^{6}-z^{2}a^{6}-z^{4}a^{4}-z^{2}a^{4}-a^{4}+2z^{2}a^{2}+2a^{2}$
Kauffman polynomial (db, data sources)	$z^{6}a^{10}-3z^{4}a^{10}+z^{2}a^{10}+3z^{7}a^{9}-11z^{5}a^{9}+8z^{3}a^{9}-za^{9}+3z^{8}a^{8}-11z^{6}a^{8}+9z^{4}a^{8}-2z^{2}a^{8}+z^{9}a^{7}-10z^{5}a^{7}+11z^{3}a^{7}-3za^{7}+4z^{8}a^{6}-16z^{6}a^{6}+19z^{4}a^{6}-6z^{2}a^{6}+z^{9}a^{5}-3z^{7}a^{5}+2z^{5}a^{5}+3z^{3}a^{5}-2za^{5}+z^{8}a^{4}-4z^{6}a^{4}+7z^{4}a^{4}-z^{2}a^{4}-a^{4}+z^{5}a^{3}+2z^{2}a^{2}-2a^{2}$
The A2 invariant	Data:K11n84/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n84/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n84 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["K11n84"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{2}+9t-13+9t^{-1}-2t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 35, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{2}a^{8}-z^{4}a^{6}-z^{2}a^{6}-z^{4}a^{4}-z^{2}a^{4}-a^{4}+2z^{2}a^{2}+2a^{2}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{6}a^{10}-3z^{4}a^{10}+z^{2}a^{10}+3z^{7}a^{9}-11z^{5}a^{9}+8z^{3}a^{9}-za^{9}+3z^{8}a^{8}-11z^{6}a^{8}+9z^{4}a^{8}-2z^{2}a^{8}+z^{9}a^{7}-10z^{5}a^{7}+11z^{3}a^{7}-3za^{7}+4z^{8}a^{6}-16z^{6}a^{6}+19z^{4}a^{6}-6z^{2}a^{6}+z^{9}a^{5}-3z^{7}a^{5}+2z^{5}a^{5}+3z^{3}a^{5}-2za^{5}+z^{8}a^{4}-4z^{6}a^{4}+7z^{4}a^{4}-z^{2}a^{4}-a^{4}+z^{5}a^{3}+2z^{2}a^{2}-2a^{2}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_12,}

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["K11n84"];

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]=

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

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]=

{9_12,}

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:

(1, -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 $4$ $-8$ $8$ ${\frac {110}{3}}$ ${\frac {58}{3}}$ $-32$ $-{\frac {464}{3}}$ $-{\frac {224}{3}}$ $-40$ ${\frac {32}{3}}$ $32$ ${\frac {440}{3}}$ ${\frac {232}{3}}$ ${\frac {15871}{30}}$ ${\frac {458}{15}}$ ${\frac {17942}{45}}$ ${\frac {65}{18}}$ ${\frac {1951}{30}}$

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 $s=$-2 is the signature of K11n84. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -8 -7 -6 -5 -4 -3 -2 -1 0 χ -1                 2 2 -3               2 1 -1 -5             3 1   2 -7           3 2     -1 -9         3 3       0 -11       2 3         1 -13     2 3           -1 -15   1 2             1 -17   2               -2 -19 1                 1

Integral Khovanov Homology (db, data source)

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

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.

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