K11a273

K11a272

K11a274

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a273 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X10,3,11,4 X14,5,15,6 X16,8,17,7 X20,9,21,10 X4,11,5,12 X18,13,19,14 X2,15,3,16 X22,18,1,17 X12,19,13,20 X8,21,9,22
Gauss code	1, -8, 2, -6, 3, -1, 4, -11, 5, -2, 6, -10, 7, -3, 8, -4, 9, -7, 10, -5, 11, -9
Dowker-Thistlethwaite code	6 10 14 16 20 4 18 2 22 12 8

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a273/ThurstonBennequinNumber Hyperbolic Volume 18.4008 A-Polynomial See Data:K11a273/A-polynomial

[edit Notes for K11a273's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant 2

[edit Notes for K11a273's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$3t^{3}-16t^{2}+37t-47+37t^{-1}-16t^{-2}+3t^{-3}$
Conway polynomial	$3z^{6}+2z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 159, -2 }
Jones polynomial	$-q^{2}+4q-9+17q^{-1}-22q^{-2}+26q^{-3}-26q^{-4}+22q^{-5}-17q^{-6}+10q^{-7}-4q^{-8}+q^{-9}$
HOMFLY-PT polynomial (db, data sources)	$z^{2}a^{8}+a^{8}-3z^{4}a^{6}-5z^{2}a^{6}-2a^{6}+2z^{6}a^{4}+5z^{4}a^{4}+4z^{2}a^{4}+z^{6}a^{2}+z^{4}a^{2}+z^{2}a^{2}+2a^{2}-z^{4}-z^{2}$
Kauffman polynomial (db, data sources)	$z^{6}a^{10}-2z^{4}a^{10}+z^{2}a^{10}+4z^{7}a^{9}-8z^{5}a^{9}+5z^{3}a^{9}-za^{9}+8z^{8}a^{8}-17z^{6}a^{8}+12z^{4}a^{8}-5z^{2}a^{8}+a^{8}+8z^{9}a^{7}-11z^{7}a^{7}-z^{5}a^{7}+z^{3}a^{7}+za^{7}+3z^{10}a^{6}+14z^{8}a^{6}-45z^{6}a^{6}+39z^{4}a^{6}-16z^{2}a^{6}+2a^{6}+17z^{9}a^{5}-30z^{7}a^{5}+15z^{5}a^{5}-8z^{3}a^{5}+5za^{5}+3z^{10}a^{4}+17z^{8}a^{4}-47z^{6}a^{4}+40z^{4}a^{4}-13z^{2}a^{4}+9z^{9}a^{3}-7z^{7}a^{3}-3z^{5}a^{3}+z^{3}a^{3}+3za^{3}+11z^{8}a^{2}-16z^{6}a^{2}+10z^{4}a^{2}-z^{2}a^{2}-2a^{2}+8z^{7}a-10z^{5}a+4z^{3}a+4z^{6}-5z^{4}+2z^{2}+z^{5}a^{-1}-z^{3}a^{-1}$
The A2 invariant	Data:K11a273/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a273/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a273 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["K11a273"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$3t^{3}-16t^{2}+37t-47+37t^{-1}-16t^{-2}+3t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$3z^{6}+2z^{4}+1$

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]=

{ 159, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{2}+4q-9+17q^{-1}-22q^{-2}+26q^{-3}-26q^{-4}+22q^{-5}-17q^{-6}+10q^{-7}-4q^{-8}+q^{-9}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{2}a^{8}+a^{8}-3z^{4}a^{6}-5z^{2}a^{6}-2a^{6}+2z^{6}a^{4}+5z^{4}a^{4}+4z^{2}a^{4}+z^{6}a^{2}+z^{4}a^{2}+z^{2}a^{2}+2a^{2}-z^{4}-z^{2}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{6}a^{10}-2z^{4}a^{10}+z^{2}a^{10}+4z^{7}a^{9}-8z^{5}a^{9}+5z^{3}a^{9}-za^{9}+8z^{8}a^{8}-17z^{6}a^{8}+12z^{4}a^{8}-5z^{2}a^{8}+a^{8}+8z^{9}a^{7}-11z^{7}a^{7}-z^{5}a^{7}+z^{3}a^{7}+za^{7}+3z^{10}a^{6}+14z^{8}a^{6}-45z^{6}a^{6}+39z^{4}a^{6}-16z^{2}a^{6}+2a^{6}+17z^{9}a^{5}-30z^{7}a^{5}+15z^{5}a^{5}-8z^{3}a^{5}+5za^{5}+3z^{10}a^{4}+17z^{8}a^{4}-47z^{6}a^{4}+40z^{4}a^{4}-13z^{2}a^{4}+9z^{9}a^{3}-7z^{7}a^{3}-3z^{5}a^{3}+z^{3}a^{3}+3za^{3}+11z^{8}a^{2}-16z^{6}a^{2}+10z^{4}a^{2}-z^{2}a^{2}-2a^{2}+8z^{7}a-10z^{5}a+4z^{3}a+4z^{6}-5z^{4}+2z^{2}+z^{5}a^{-1}-z^{3}a^{-1}$

"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["K11a273"];

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]=

{ $3t^{3}-16t^{2}+37t-47+37t^{-1}-16t^{-2}+3t^{-3}$, $-q^{2}+4q-9+17q^{-1}-22q^{-2}+26q^{-3}-26q^{-4}+22q^{-5}-17q^{-6}+10q^{-7}-4q^{-8}+q^{-9}$ }

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]=

{}

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:

(0, 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 $0$ $16$ $0$ $-80$ $-16$ $0$ ${\frac {640}{3}}$ $-{\frac {32}{3}}$ $80$ $0$ $128$ $0$ $0$ $-136$ $144$ $-{\frac {224}{3}}$ $-{\frac {296}{3}}$ $8$

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 K11a273. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 5                       1 -1 3                     3   3 1                   6 1   -5 -1                 11 3     8 -3               12 7       -5 -5             14 10         4 -7           12 12           0 -9         10 14             -4 -11       7 12               5 -13     3 10                 -7 -15   1 7                   6 -17   3                     -3 -19 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-8$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-5$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-4$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-3$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=-2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{14}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=0$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{11}$ $r=1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=3$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a272 K11a274