K11n49

K11n48

K11n50

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n49 at Knotilus!

K11n49 is not $k$-colourable for any $k$. See The Determinant and the Signature.

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n49/ThurstonBennequinNumber Hyperbolic Volume 6.90911 A-Polynomial See Data:K11n49/A-polynomial

[edit Notes for K11n49'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 K11n49's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{2}+3-t^{-2}$
Conway polynomial	$-z^{4}-4z^{2}+1$
2nd Alexander ideal (db, data sources)	$\left\{2,t^{2}+t+1\right\}$
Determinant and Signature	{ 1, 0 }
Jones polynomial	$q^{4}-q^{3}+q^{2}-q+1-q^{-2}+q^{-3}-q^{-4}+q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$z^{2}a^{4}+2a^{4}-z^{4}a^{2}-4z^{2}a^{2}-3a^{2}+2-z^{2}a^{-2}-a^{-2}+a^{-4}$
Kauffman polynomial (db, data sources)	$a^{3}z^{9}+az^{9}+a^{4}z^{8}+2a^{2}z^{8}+z^{8}-7a^{3}z^{7}-7az^{7}-7a^{4}z^{6}-14a^{2}z^{6}-7z^{6}+15a^{3}z^{5}+14az^{5}+z^{5}a^{-3}+15a^{4}z^{4}+28a^{2}z^{4}+z^{4}a^{-2}+z^{4}a^{-4}+13z^{4}-12a^{3}z^{3}-10az^{3}-z^{3}a^{-1}-3z^{3}a^{-3}-11a^{4}z^{2}-18a^{2}z^{2}-3z^{2}a^{-2}-3z^{2}a^{-4}-7z^{2}+3a^{3}z+3az+za^{-1}+za^{-3}+2a^{4}+3a^{2}+a^{-2}+a^{-4}+2$
The A2 invariant	Data:K11n49/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n49/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n49 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["K11n49"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

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

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 1, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{3}z^{9}+az^{9}+a^{4}z^{8}+2a^{2}z^{8}+z^{8}-7a^{3}z^{7}-7az^{7}-7a^{4}z^{6}-14a^{2}z^{6}-7z^{6}+15a^{3}z^{5}+14az^{5}+z^{5}a^{-3}+15a^{4}z^{4}+28a^{2}z^{4}+z^{4}a^{-2}+z^{4}a^{-4}+13z^{4}-12a^{3}z^{3}-10az^{3}-z^{3}a^{-1}-3z^{3}a^{-3}-11a^{4}z^{2}-18a^{2}z^{2}-3z^{2}a^{-2}-3z^{2}a^{-4}-7z^{2}+3a^{3}z+3az+za^{-1}+za^{-3}+2a^{4}+3a^{2}+a^{-2}+a^{-4}+2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n116,}

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["K11n49"];

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]=

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

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]=

{K11n116,}

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:

(-4, 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 $-16$ $8$ $128$ ${\frac {520}{3}}$ ${\frac {224}{3}}$ $-128$ $-{\frac {688}{3}}$ $-{\frac {160}{3}}$ $-56$ $-{\frac {2048}{3}}$ $32$ $-{\frac {8320}{3}}$ $-{\frac {3584}{3}}$ $-{\frac {34862}{15}}$ ${\frac {2576}{5}}$ $-{\frac {102848}{45}}$ ${\frac {2942}{9}}$ $-{\frac {7982}{15}}$

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 K11n49. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11n48 K11n50