K11n21

K11n20

K11n22

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n21 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X12,5,13,6 X2837 X9,15,10,14 X11,19,12,18 X6,13,7,14 X15,20,16,21 X17,11,18,10 X19,22,20,1 X21,16,22,17
Gauss code	1, -4, 2, -1, 3, -7, 4, -2, -5, 9, -6, -3, 7, 5, -8, 11, -9, 6, -10, 8, -11, 10
Dowker-Thistlethwaite code	4 8 12 2 -14 -18 6 -20 -10 -22 -16

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n21/ThurstonBennequinNumber Hyperbolic Volume 12.468 A-Polynomial See Data:K11n21/A-polynomial

[edit Notes for K11n21'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 K11n21's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+5t^{2}-11t+15-11t^{-1}+5t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}-z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 49, 0 }
Jones polynomial	$q^{6}-3q^{5}+5q^{4}-7q^{3}+8q^{2}-8q+8-5q^{-1}+3q^{-2}-q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-2}-4z^{4}a^{-2}+z^{4}a^{-4}+2z^{4}-a^{2}z^{2}-6z^{2}a^{-2}+2z^{2}a^{-4}+5z^{2}-a^{2}-3a^{-2}+a^{-4}+4$
Kauffman polynomial (db, data sources)	$z^{9}a^{-1}+z^{9}a^{-3}+5z^{8}a^{-2}+3z^{8}a^{-4}+2z^{8}+az^{7}+z^{7}a^{-1}+3z^{7}a^{-3}+3z^{7}a^{-5}-15z^{6}a^{-2}-8z^{6}a^{-4}+z^{6}a^{-6}-6z^{6}-az^{5}-8z^{5}a^{-1}-17z^{5}a^{-3}-10z^{5}a^{-5}+3a^{2}z^{4}+16z^{4}a^{-2}+3z^{4}a^{-4}-3z^{4}a^{-6}+13z^{4}+a^{3}z^{3}+5az^{3}+12z^{3}a^{-1}+16z^{3}a^{-3}+8z^{3}a^{-5}-3a^{2}z^{2}-10z^{2}a^{-2}-z^{2}a^{-4}+2z^{2}a^{-6}-10z^{2}-a^{3}z-3az-5za^{-1}-5za^{-3}-2za^{-5}+a^{2}+3a^{-2}+a^{-4}+4$
The A2 invariant	$-q^{10}+q^{6}-q^{4}+3q^{2}+1+q^{-2}+q^{-4}-2q^{-6}+q^{-8}-2q^{-10}+q^{-14}-q^{-16}+q^{-18}$
The G2 invariant	Data:K11n21/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n21 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["K11n21"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+5t^{2}-11t+15-11t^{-1}+5t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 49, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{9}a^{-1}+z^{9}a^{-3}+5z^{8}a^{-2}+3z^{8}a^{-4}+2z^{8}+az^{7}+z^{7}a^{-1}+3z^{7}a^{-3}+3z^{7}a^{-5}-15z^{6}a^{-2}-8z^{6}a^{-4}+z^{6}a^{-6}-6z^{6}-az^{5}-8z^{5}a^{-1}-17z^{5}a^{-3}-10z^{5}a^{-5}+3a^{2}z^{4}+16z^{4}a^{-2}+3z^{4}a^{-4}-3z^{4}a^{-6}+13z^{4}+a^{3}z^{3}+5az^{3}+12z^{3}a^{-1}+16z^{3}a^{-3}+8z^{3}a^{-5}-3a^{2}z^{2}-10z^{2}a^{-2}-z^{2}a^{-4}+2z^{2}a^{-6}-10z^{2}-a^{3}z-3az-5za^{-1}-5za^{-3}-2za^{-5}+a^{2}+3a^{-2}+a^{-4}+4$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_27, K11n4, K11n172,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {9_41, K11n4,}

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["K11n21"];

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]=

{ $-t^{3}+5t^{2}-11t+15-11t^{-1}+5t^{-2}-t^{-3}$, $q^{6}-3q^{5}+5q^{4}-7q^{3}+8q^{2}-8q+8-5q^{-1}+3q^{-2}-q^{-3}$ }

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_27, K11n4, K11n172,}

In[6]:=

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

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

Out[6]=

{9_41, K11n4,}

Vassiliev invariants

V2 and V3:

(0, -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 $0$ $-8$ $0$ $0$ $8$ $0$ ${\frac {112}{3}}$ ${\frac {64}{3}}$ $24$ $0$ $32$ $0$ $0$ $112$ ${\frac {200}{3}}$ $-{\frac {8}{3}}$ ${\frac {64}{3}}$ $-16$

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 K11n21. 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 13                   1 1 11                 2   -2 9               3 1   2 7             4 2     -2 5           4 3       1 3         4 4         0 1       4 4           0 -1     2 5             3 -3   1 3               -2 -5   2                 2 -7 1                   -1

Integral Khovanov Homology (db, data source)

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

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.

K11n20 K11n22