K11a4

K11a3

K11a5

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a4 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8394 X10,6,11,5 X14,8,15,7 X2,9,3,10 X18,11,19,12 X6,14,7,13 X22,15,1,16 X20,17,21,18 X12,19,13,20 X16,21,17,22
Gauss code	1, -5, 2, -1, 3, -7, 4, -2, 5, -3, 6, -10, 7, -4, 8, -11, 9, -6, 10, -9, 11, -8
Dowker-Thistlethwaite code	4 8 10 14 2 18 6 22 20 12 16

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a4/ThurstonBennequinNumber Hyperbolic Volume 14.1846 A-Polynomial See Data:K11a4/A-polynomial

[edit Notes for K11a4's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant 0

[edit Notes for K11a4's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+10t^{2}-22t+29-22t^{-1}+10t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}-2z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 97, 0 }
Jones polynomial	$q^{4}-3q^{3}+6q^{2}-10q+14-15q^{-1}+15q^{-2}-13q^{-3}+10q^{-4}-6q^{-5}+3q^{-6}-q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$-z^{2}a^{6}-a^{6}+2z^{4}a^{4}+4z^{2}a^{4}+2a^{4}-z^{6}a^{2}-2z^{4}a^{2}-z^{2}a^{2}-z^{6}-3z^{4}-4z^{2}-1+z^{4}a^{-2}+2z^{2}a^{-2}+a^{-2}$
Kauffman polynomial (db, data sources)	$a^{4}z^{10}+a^{2}z^{10}+3a^{5}z^{9}+7a^{3}z^{9}+4az^{9}+3a^{6}z^{8}+7a^{4}z^{8}+10a^{2}z^{8}+6z^{8}+a^{7}z^{7}-7a^{5}z^{7}-15a^{3}z^{7}-az^{7}+6z^{7}a^{-1}-12a^{6}z^{6}-34a^{4}z^{6}-33a^{2}z^{6}+5z^{6}a^{-2}-6z^{6}-4a^{7}z^{5}-3a^{5}z^{5}-a^{3}z^{5}-10az^{5}-5z^{5}a^{-1}+3z^{5}a^{-3}+15a^{6}z^{4}+39a^{4}z^{4}+28a^{2}z^{4}-5z^{4}a^{-2}+z^{4}a^{-4}-2z^{4}+5a^{7}z^{3}+11a^{5}z^{3}+10a^{3}z^{3}+7az^{3}-3z^{3}a^{-3}-7a^{6}z^{2}-16a^{4}z^{2}-8a^{2}z^{2}+3z^{2}a^{-2}-z^{2}a^{-4}+5z^{2}-2a^{7}z-4a^{5}z-3a^{3}z-az+za^{-1}+za^{-3}+a^{6}+2a^{4}-a^{-2}-1$
The A2 invariant	Data:K11a4/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a4/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a4 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["K11a4"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{3}+10t^{2}-22t+29-22t^{-1}+10t^{-2}-2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 97, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{4}-3q^{3}+6q^{2}-10q+14-15q^{-1}+15q^{-2}-13q^{-3}+10q^{-4}-6q^{-5}+3q^{-6}-q^{-7}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{4}z^{10}+a^{2}z^{10}+3a^{5}z^{9}+7a^{3}z^{9}+4az^{9}+3a^{6}z^{8}+7a^{4}z^{8}+10a^{2}z^{8}+6z^{8}+a^{7}z^{7}-7a^{5}z^{7}-15a^{3}z^{7}-az^{7}+6z^{7}a^{-1}-12a^{6}z^{6}-34a^{4}z^{6}-33a^{2}z^{6}+5z^{6}a^{-2}-6z^{6}-4a^{7}z^{5}-3a^{5}z^{5}-a^{3}z^{5}-10az^{5}-5z^{5}a^{-1}+3z^{5}a^{-3}+15a^{6}z^{4}+39a^{4}z^{4}+28a^{2}z^{4}-5z^{4}a^{-2}+z^{4}a^{-4}-2z^{4}+5a^{7}z^{3}+11a^{5}z^{3}+10a^{3}z^{3}+7az^{3}-3z^{3}a^{-3}-7a^{6}z^{2}-16a^{4}z^{2}-8a^{2}z^{2}+3z^{2}a^{-2}-z^{2}a^{-4}+5z^{2}-2a^{7}z-4a^{5}z-3a^{3}z-az+za^{-1}+za^{-3}+a^{6}+2a^{4}-a^{-2}-1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a110,}

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["K11a4"];

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]=

{ $-2t^{3}+10t^{2}-22t+29-22t^{-1}+10t^{-2}-2t^{-3}$, $q^{4}-3q^{3}+6q^{2}-10q+14-15q^{-1}+15q^{-2}-13q^{-3}+10q^{-4}-6q^{-5}+3q^{-6}-q^{-7}$ }

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]=

{K11a110,}

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 $0$ $-16$ $0$ $48$ $16$ $0$ $-{\frac {64}{3}}$ ${\frac {32}{3}}$ $16$ $0$ $128$ $0$ $0$ $216$ ${\frac {352}{3}}$ $-{\frac {352}{3}}$ ${\frac {104}{3}}$ $-40$

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 K11a4. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11a3 K11a5