© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11n93 K11n95

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,4,11,3 X14,6,15,5 X12,7,13,8 X2,10,3,9 X20,11,21,12 X18,14,19,13 X15,22,16,1 X6,18,7,17 X8,19,9,20 X21,16,22,17

Gauss Code:

{1, -5, 2, -1, 3, -9, 4, -10, 5, -2, 6, -4, 7, -3, -8, 11, 9, -7, 10, -6, -11, 8}

DT (Dowker-Thistlethwaite) Code:

4 10 14 12 2 20 18 -22 6 8 -16

Alexander Polynomial:

- t-3 + 6t-2 - 13t-1 + 17 - 13t + 6t2 - t3

Conway Polynomial:

1 + 2z2 - z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{57, 0}

Jones Polynomial:

- q-3 + 3q-2 - 6q-1 + 9 - 9q + 10q2 - 8q3 + 6q4 - 4q5 + q6

Other knots (up to mirrors) with the same Jones Polynomial:

{...}

A2 (sl(3)) Invariant:

- q-10 + q-6 - 2q-4 + 2q-2 + q2 + 3q4 + 3q8 - 2q10 - q12 - 2q16 + q18

HOMFLY-PT Polynomial:

- a-4 + a-4z2 + a-4z4 + a-2 - 2a-2z2 - 3a-2z4 - a-2z6 + 2 + 4z2 + 2z4 - a2 - a2z2

Kauffman Polynomial:

- 2a-6z4 + a-6z6 + a-5z + 6a-5z3 - 12a-5z5 + 4a-5z7 - a-4 - 3a-4z2 + 12a-4z4 - 16a-4z6 + 5a-4z8 + 11a-3z3 - 14a-3z5 + 2a-3z9 - a-2 - 10a-2z2 + 29a-2z4 - 27a-2z6 + 8a-2z8 - 2a-1z + 7a-1z3 - 2a-1z5 - 3a-1z7 + 2a-1z9 + 2 - 10z2 + 18z4 - 10z6 + 3z8 - 2az + 3az3 + az7 + a2 - 3a2z2 + 3a2z4 - a3z + a3z3

V2 and V3, the type 2 and 3 Vassiliev invariants:

{2, 1}

Khovanov Homology: (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 1194. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 13                   1 j = 11                 3   j = 9               3 1   j = 7             5 3     j = 5           5 3       j = 3         4 5         j = 1       5 5           j = -1     2 5             j = -3   1 4               j = -5   2                 j = -7 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Show[DrawMorseLink[Knot[11, NonAlternating, 94]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 94]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[14, 6, 15, 5], X[12, 7, 13, 8], > X[2, 10, 3, 9], X[20, 11, 21, 12], X[18, 14, 19, 13], X[15, 22, 16, 1], > X[6, 18, 7, 17], X[8, 19, 9, 20], X[21, 16, 22, 17]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 94]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -9, 4, -10, 5, -2, 6, -4, 7, -3, -8, 11, 9, -7, 10, > -6, -11, 8]
In[5]:=	DTCode[Knot[11, NonAlternating, 94]]
Out[5]=	DTCode[4, 10, 14, 12, 2, 20, 18, -22, 6, 8, -16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 94]][t]
Out[6]=	-3 6 13 2 3 17 - t + -- - -- - 13 t + 6 t - t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 94]][z]
Out[7]=	2 6 1 + 2 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 94]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 94]], KnotSignature[Knot[11, NonAlternating, 94]]}
Out[9]=	{57, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 94]][q]
Out[10]=	-3 3 6 2 3 4 5 6 9 - q + -- - - - 9 q + 10 q - 8 q + 6 q - 4 q + q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 94]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 94]][q]
Out[12]=	-10 -6 2 2 2 4 8 10 12 16 18 -q + q - -- + -- + q + 3 q + 3 q - 2 q - q - 2 q + q 4 2 q q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 94]][a, z]
Out[13]=	2 2 4 4 6 -4 -2 2 2 z 2 z 2 2 4 z 3 z z 2 - a + a - a + 4 z + -- - ---- - a z + 2 z + -- - ---- - -- 4 2 4 2 2 a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 94]][a, z]
Out[14]=	2 2 -4 -2 2 z 2 z 3 2 3 z 10 z 2 2 2 - a - a + a + -- - --- - 2 a z - a z - 10 z - ---- - ----- - 3 a z + 5 a 4 2 a a a 3 3 3 4 4 4 6 z 11 z 7 z 3 3 3 4 2 z 12 z 29 z > ---- + ----- + ---- + 3 a z + a z + 18 z - ---- + ----- + ----- + 5 3 a 6 4 2 a a a a a 5 5 5 6 6 6 7 7 2 4 12 z 14 z 2 z 6 z 16 z 27 z 4 z 3 z > 3 a z - ----- - ----- - ---- - 10 z + -- - ----- - ----- + ---- - ---- + 5 3 a 6 4 2 5 a a a a a a a 8 8 9 9 7 8 5 z 8 z 2 z 2 z > a z + 3 z + ---- + ---- + ---- + ---- 4 2 3 a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 94]], Vassiliev[3][Knot[11, NonAlternating, 94]]}
Out[15]=	{2, 1}
In[16]:=	Kh[Knot[11, NonAlternating, 94]][q, t]
Out[16]=	5 1 2 1 4 2 3 3 2 - + 5 q + ----- + ----- + ----- + ---- + --- + 5 q t + 4 q t + 5 q t + q 7 3 5 2 3 2 3 q t q t q t q t q t 5 2 5 3 7 3 7 4 9 4 9 5 11 5 13 6 > 5 q t + 3 q t + 5 q t + 3 q t + 3 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n94

K11n93 K11n95