K11a94

K11a93

K11a95

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	K11a94 Quick Notes

K11a94 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X10,4,11,3 X12,6,13,5 X18,8,19,7 X2,10,3,9 X8,12,9,11 X20,14,21,13 X22,16,1,15 X6,18,7,17 X16,20,17,19 X14,22,15,21
Gauss code	1, -5, 2, -1, 3, -9, 4, -6, 5, -2, 6, -3, 7, -11, 8, -10, 9, -4, 10, -7, 11, -8
Dowker-Thistlethwaite code	4 10 12 18 2 8 20 22 6 16 14
Conway Notation	[221,3,2+]

Three dimensional invariants

Symmetry type Reversible Unknotting number 3 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a94/ThurstonBennequinNumber Hyperbolic Volume 14.458 A-Polynomial See Data:K11a94/A-polynomial

[edit Notes for K11a94's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant -6

[edit Notes for K11a94's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$4t^{3}-13t^{2}+23t-27+23t^{-1}-13t^{-2}+4t^{-3}$
Conway polynomial	$4z^{6}+11z^{4}+7z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 107, 6 }
Jones polynomial	$-q^{14}+4q^{13}-8q^{12}+12q^{11}-16q^{10}+17q^{9}-17q^{8}+14q^{7}-9q^{6}+6q^{5}-2q^{4}+q^{3}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-6}+2z^{6}a^{-8}+z^{6}a^{-10}+4z^{4}a^{-6}+7z^{4}a^{-8}+z^{4}a^{-10}-z^{4}a^{-12}+5z^{2}a^{-6}+7z^{2}a^{-8}-4z^{2}a^{-10}-z^{2}a^{-12}+2a^{-6}+2a^{-8}-4a^{-10}+a^{-12}$
Kauffman polynomial (db, data sources)	$z^{10}a^{-10}+z^{10}a^{-12}+2z^{9}a^{-9}+6z^{9}a^{-11}+4z^{9}a^{-13}+3z^{8}a^{-8}+5z^{8}a^{-10}+9z^{8}a^{-12}+7z^{8}a^{-14}+2z^{7}a^{-7}+2z^{7}a^{-9}-6z^{7}a^{-11}+z^{7}a^{-13}+7z^{7}a^{-15}+z^{6}a^{-6}-7z^{6}a^{-8}-13z^{6}a^{-10}-19z^{6}a^{-12}-10z^{6}a^{-14}+4z^{6}a^{-16}-5z^{5}a^{-7}-14z^{5}a^{-9}-7z^{5}a^{-11}-11z^{5}a^{-13}-12z^{5}a^{-15}+z^{5}a^{-17}-4z^{4}a^{-6}+6z^{4}a^{-8}+13z^{4}a^{-10}+12z^{4}a^{-12}+3z^{4}a^{-14}-6z^{4}a^{-16}+2z^{3}a^{-7}+15z^{3}a^{-9}+17z^{3}a^{-11}+10z^{3}a^{-13}+5z^{3}a^{-15}-z^{3}a^{-17}+5z^{2}a^{-6}-5z^{2}a^{-8}-10z^{2}a^{-10}-z^{2}a^{-12}+z^{2}a^{-16}+za^{-7}-6za^{-9}-9za^{-11}-3za^{-13}-za^{-15}-2a^{-6}+2a^{-8}+4a^{-10}+a^{-12}$
The A2 invariant	Data:K11a94/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a94/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a94 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["K11a94"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$4t^{3}-13t^{2}+23t-27+23t^{-1}-13t^{-2}+4t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$4z^{6}+11z^{4}+7z^{2}+1$

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]=

{ 107, 6 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{14}+4q^{13}-8q^{12}+12q^{11}-16q^{10}+17q^{9}-17q^{8}+14q^{7}-9q^{6}+6q^{5}-2q^{4}+q^{3}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{6}a^{-6}+2z^{6}a^{-8}+z^{6}a^{-10}+4z^{4}a^{-6}+7z^{4}a^{-8}+z^{4}a^{-10}-z^{4}a^{-12}+5z^{2}a^{-6}+7z^{2}a^{-8}-4z^{2}a^{-10}-z^{2}a^{-12}+2a^{-6}+2a^{-8}-4a^{-10}+a^{-12}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-10}+z^{10}a^{-12}+2z^{9}a^{-9}+6z^{9}a^{-11}+4z^{9}a^{-13}+3z^{8}a^{-8}+5z^{8}a^{-10}+9z^{8}a^{-12}+7z^{8}a^{-14}+2z^{7}a^{-7}+2z^{7}a^{-9}-6z^{7}a^{-11}+z^{7}a^{-13}+7z^{7}a^{-15}+z^{6}a^{-6}-7z^{6}a^{-8}-13z^{6}a^{-10}-19z^{6}a^{-12}-10z^{6}a^{-14}+4z^{6}a^{-16}-5z^{5}a^{-7}-14z^{5}a^{-9}-7z^{5}a^{-11}-11z^{5}a^{-13}-12z^{5}a^{-15}+z^{5}a^{-17}-4z^{4}a^{-6}+6z^{4}a^{-8}+13z^{4}a^{-10}+12z^{4}a^{-12}+3z^{4}a^{-14}-6z^{4}a^{-16}+2z^{3}a^{-7}+15z^{3}a^{-9}+17z^{3}a^{-11}+10z^{3}a^{-13}+5z^{3}a^{-15}-z^{3}a^{-17}+5z^{2}a^{-6}-5z^{2}a^{-8}-10z^{2}a^{-10}-z^{2}a^{-12}+z^{2}a^{-16}+za^{-7}-6za^{-9}-9za^{-11}-3za^{-13}-za^{-15}-2a^{-6}+2a^{-8}+4a^{-10}+a^{-12}$

Vassiliev invariants

V2 and V3:

(7, 17)

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 $28$ $136$ $392$ ${\frac {2546}{3}}$ ${\frac {310}{3}}$ $3808$ ${\frac {17872}{3}}$ ${\frac {2848}{3}}$ $616$ ${\frac {10976}{3}}$ $9248$ ${\frac {71288}{3}}$ ${\frac {8680}{3}}$ ${\frac {1301737}{30}}$ ${\frac {13682}{5}}$ ${\frac {600554}{45}}$ ${\frac {5399}{18}}$ ${\frac {46537}{30}}$

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=$6 is the signature of K11a94. 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 10 11 χ 29                       1 -1 27                     3   3 25                   5 1   -4 23                 7 3     4 21               9 5       -4 19             8 7         1 17           9 9           0 15         5 8             -3 13       4 9               5 11     2 5                 -3 9     4                   4 7 1 2                     -1 5 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=5$ $i=7$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=1$ ${\mathbb {Z} }^{2}$ $r=2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=4$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=5$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=6$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=7$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=8$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=9$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=10$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=11$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[11, Alternating, 94]]
Out[2]=	11
In[3]:=	PD[Knot[11, Alternating, 94]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[12, 6, 13, 5], X[18, 8, 19, 7], X[2, 10, 3, 9], X[8, 12, 9, 11], X[20, 14, 21, 13], X[22, 16, 1, 15], X[6, 18, 7, 17], X[16, 20, 17, 19], X[14, 22, 15, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 94]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -9, 4, -6, 5, -2, 6, -3, 7, -11, 8, -10, 9, -4, 10, -7, 11, -8]
In[5]:=	BR[Knot[11, Alternating, 94]]
Out[5]=	BR[Knot[11, Alternating, 94]]
In[6]:=	alex = Alexander[Knot[11, Alternating, 94]][t]
Out[6]=	4 13 23 2 3 -27 + -- - -- + -- + 23 t - 13 t + 4 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 94]][z]
Out[7]=	2 4 6 1 + 7 z + 11 z + 4 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 94]}
In[9]:=	{KnotDet[Knot[11, Alternating, 94]], KnotSignature[Knot[11, Alternating, 94]]}
Out[9]=	{107, 6}
In[10]:=	J=Jones[Knot[11, Alternating, 94]][q]
Out[10]=	3 4 5 6 7 8 9 10 11 q - 2 q + 6 q - 9 q + 14 q - 17 q + 17 q - 16 q + 12 q - 12 13 14 8 q + 4 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 94]}
In[12]:=	A2Invariant[Knot[11, Alternating, 94]][q]
Out[12]=	10 12 14 18 20 22 24 26 28 q - q + 3 q + q + 4 q - 2 q + 3 q - 2 q - 2 q - 32 34 36 40 42 4 q + 2 q - q + 2 q - q
In[13]:=	Kauffman[Knot[11, Alternating, 94]][a, z]
Out[13]=	2 2 2 -12 4 2 2 z 3 z 9 z 6 z z z z 10 z a + --- + -- - -- - --- - --- - --- - --- + -- + --- - --- - ----- - 10 8 6 15 13 11 9 7 16 12 10 a a a a a a a a a a a 2 2 3 3 3 3 3 3 4 5 z 5 z z 5 z 10 z 17 z 15 z 2 z 6 z ---- + ---- - --- + ---- + ----- + ----- + ----- + ---- - ---- + 8 6 17 15 13 11 9 7 16 a a a a a a a a a 4 4 4 4 4 5 5 5 5 3 z 12 z 13 z 6 z 4 z z 12 z 11 z 7 z ---- + ----- + ----- + ---- - ---- + --- - ----- - ----- - ---- - 14 12 10 8 6 17 15 13 11 a a a a a a a a a 5 5 6 6 6 6 6 6 7 14 z 5 z 4 z 10 z 19 z 13 z 7 z z 7 z ----- - ---- + ---- - ----- - ----- - ----- - ---- + -- + ---- + 9 7 16 14 12 10 8 6 15 a a a a a a a a a 7 7 7 7 8 8 8 8 9 9 z 6 z 2 z 2 z 7 z 9 z 5 z 3 z 4 z 6 z --- - ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + 13 11 9 7 14 12 10 8 13 11 a a a a a a a a a a 9 10 10 2 z z z ---- + --- + --- 9 12 10 a a a
In[14]:=	{Vassiliev[2][Knot[11, Alternating, 94]], Vassiliev[3][Knot[11, Alternating, 94]]}
Out[14]=	{0, 17}
In[15]:=	Kh[Knot[11, Alternating, 94]][q, t]
Out[15]=	5 7 7 9 2 11 2 11 3 13 3 q + q + 2 q t + 4 q t + 2 q t + 5 q t + 4 q t + 13 4 15 4 15 5 17 5 17 6 19 6 9 q t + 5 q t + 8 q t + 9 q t + 9 q t + 8 q t + 19 7 21 7 21 8 23 8 23 9 25 9 7 q t + 9 q t + 5 q t + 7 q t + 3 q t + 5 q t + 25 10 27 10 29 11 q t + 3 q t + q t