K11n105

K11n104

K11n106

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n105 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,4,11,3 X14,5,15,6 X16,8,17,7 X2,10,3,9 X11,21,12,20 X13,1,14,22 X18,16,19,15 X8,18,9,17 X6,19,7,20 X21,13,22,12
Gauss code	1, -5, 2, -1, 3, -10, 4, -9, 5, -2, -6, 11, -7, -3, 8, -4, 9, -8, 10, 6, -11, 7
Dowker-Thistlethwaite code	4 10 14 16 2 -20 -22 18 8 6 -12

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n105/ThurstonBennequinNumber Hyperbolic Volume 13.6421 A-Polynomial See Data:K11n105/A-polynomial

[edit Notes for K11n105's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n105's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+7t^{2}-16t+21-16t^{-1}+7t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}+z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 69, 4 }
Jones polynomial	$-q^{11}+4q^{10}-7q^{9}+9q^{8}-12q^{7}+12q^{6}-10q^{5}+8q^{4}-4q^{3}+2q^{2}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-6}+2z^{4}a^{-4}-3z^{4}a^{-6}+2z^{4}a^{-8}+5z^{2}a^{-4}-4z^{2}a^{-6}+3z^{2}a^{-8}-z^{2}a^{-10}+3a^{-4}-2a^{-6}$
Kauffman polynomial (db, data sources)	$z^{9}a^{-7}+z^{9}a^{-9}+2z^{8}a^{-6}+6z^{8}a^{-8}+4z^{8}a^{-10}+z^{7}a^{-5}+3z^{7}a^{-7}+8z^{7}a^{-9}+6z^{7}a^{-11}-3z^{6}a^{-6}-10z^{6}a^{-8}-3z^{6}a^{-10}+4z^{6}a^{-12}+z^{5}a^{-5}-5z^{5}a^{-7}-19z^{5}a^{-9}-12z^{5}a^{-11}+z^{5}a^{-13}+3z^{4}a^{-4}+8z^{4}a^{-6}+8z^{4}a^{-8}-4z^{4}a^{-10}-7z^{4}a^{-12}-z^{3}a^{-5}+4z^{3}a^{-7}+11z^{3}a^{-9}+5z^{3}a^{-11}-z^{3}a^{-13}-6z^{2}a^{-4}-8z^{2}a^{-6}-2z^{2}a^{-8}+z^{2}a^{-10}+z^{2}a^{-12}-2za^{-5}-za^{-7}+za^{-9}+3a^{-4}+2a^{-6}$
The A2 invariant	$2q^{-6}-q^{-8}+3q^{-10}+q^{-12}-q^{-14}+3q^{-16}-2q^{-18}+q^{-20}-2q^{-22}-2q^{-24}+q^{-26}-2q^{-28}+2q^{-30}+q^{-32}-q^{-34}$
The G2 invariant	Data:K11n105/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n105 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["K11n105"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+7t^{2}-16t+21-16t^{-1}+7t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 69, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{11}+4q^{10}-7q^{9}+9q^{8}-12q^{7}+12q^{6}-10q^{5}+8q^{4}-4q^{3}+2q^{2}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{6}a^{-6}+2z^{4}a^{-4}-3z^{4}a^{-6}+2z^{4}a^{-8}+5z^{2}a^{-4}-4z^{2}a^{-6}+3z^{2}a^{-8}-z^{2}a^{-10}+3a^{-4}-2a^{-6}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{9}a^{-7}+z^{9}a^{-9}+2z^{8}a^{-6}+6z^{8}a^{-8}+4z^{8}a^{-10}+z^{7}a^{-5}+3z^{7}a^{-7}+8z^{7}a^{-9}+6z^{7}a^{-11}-3z^{6}a^{-6}-10z^{6}a^{-8}-3z^{6}a^{-10}+4z^{6}a^{-12}+z^{5}a^{-5}-5z^{5}a^{-7}-19z^{5}a^{-9}-12z^{5}a^{-11}+z^{5}a^{-13}+3z^{4}a^{-4}+8z^{4}a^{-6}+8z^{4}a^{-8}-4z^{4}a^{-10}-7z^{4}a^{-12}-z^{3}a^{-5}+4z^{3}a^{-7}+11z^{3}a^{-9}+5z^{3}a^{-11}-z^{3}a^{-13}-6z^{2}a^{-4}-8z^{2}a^{-6}-2z^{2}a^{-8}+z^{2}a^{-10}+z^{2}a^{-12}-2za^{-5}-za^{-7}+za^{-9}+3a^{-4}+2a^{-6}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_78, K11n98,}

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["K11n105"];

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]=

{ $-t^{3}+7t^{2}-16t+21-16t^{-1}+7t^{-2}-t^{-3}$, $-q^{11}+4q^{10}-7q^{9}+9q^{8}-12q^{7}+12q^{6}-10q^{5}+8q^{4}-4q^{3}+2q^{2}$ }

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]=

{10_78, K11n98,}

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:

(3, 5)

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 $12$ $40$ $72$ $174$ $26$ $480$ ${\frac {2512}{3}}$ ${\frac {352}{3}}$ $136$ $288$ $800$ $2088$ $312$ ${\frac {41151}{10}}$ $-{\frac {2306}{15}}$ ${\frac {8914}{5}}$ ${\frac {171}{2}}$ ${\frac {2271}{10}}$

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=$4 is the signature of K11n105. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ 0 1 2 3 4 5 6 7 8 9 χ 23                   1 -1 21                 3   3 19               4 1   -3 17             5 3     2 15           7 4       -3 13         5 5         0 11       5 7           2 9     3 5             -2 7   1 5               4 5 1 3                 -2 3 2                   2

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=3$ $i=5$ $r=0$ ${\mathbb {Z} }^{2}$ ${\mathbb {Z} }$ $r=1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=4$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=5$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=6$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=7$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=8$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=9$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11n104 K11n106