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

K11n95 K11n97

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,4,11,3 X5,14,6,15 X7,13,8,12 X2,10,3,9 X11,21,12,20 X13,18,14,19 X15,22,16,1 X17,6,18,7 X19,9,20,8 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 - 2t-2 + t-1 + 1 + t - 2t2 + t3

Conway Polynomial:

1 + 2z2 + 4z4 + z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{7, 2}

Jones Polynomial:

- q-4 + 2q-3 - 2q-2 + 2q-1 - 1 + q + q4 - q5

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

{...}

A2 (sl(3)) Invariant:

- q-12 + q-6 + 1 + 2q4 + q6 + q8 - q12 - q16

HOMFLY-PT Polynomial:

- a-4 - a-4z2 + a-2 + 2 + 6z2 + 5z4 + z6 - a2 - 3a2z2 - a2z4

Kauffman Polynomial:

2a-5z - 4a-5z3 + a-5z5 - a-4 + 4a-4z2 - 5a-4z4 + a-4z6 + 2a-3z - 3a-3z3 - a-2 - a-2z2 + 7a-2z4 - 6a-2z6 + a-2z8 - 2a-1z + 4a-1z3 + 3a-1z5 - 5a-1z7 + a-1z9 + 2 - 13z2 + 28z4 - 18z6 + 3z8 - 4az + 9az3 - az5 - 4az7 + az9 + a2 - 8a2z2 + 16a2z4 - 11a2z6 + 2a2z8 - 2a3z + 6a3z3 - 5a3z5 + a3z7

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=2 is the signature of 1196. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 11                     1 j = 9                       j = 7               1 1 1   j = 5             1 1       j = 3           1 1 1       j = 1         2 3 1         j = -1       1 1 1           j = -3     1 2 1             j = -5   1 1                 j = -7   1                   j = -9 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, 96]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 96]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[5, 14, 6, 15], X[7, 13, 8, 12], > X[2, 10, 3, 9], X[11, 21, 12, 20], X[13, 18, 14, 19], X[15, 22, 16, 1], > X[17, 6, 18, 7], X[19, 9, 20, 8], X[21, 16, 22, 17]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 96]]
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, 96]]
Out[5]=	DTCode[4, 10, -14, -12, 2, -20, -18, -22, -6, -8, -16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 96]][t]
Out[6]=	-3 2 1 2 3 1 + t - -- + - + t - 2 t + t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 96]][z]
Out[7]=	2 4 6 1 + 2 z + 4 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 96]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 96]], KnotSignature[Knot[11, NonAlternating, 96]]}
Out[9]=	{7, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 96]][q]
Out[10]=	-4 2 2 2 4 5 -1 - q + -- - -- + - + q + q - q 3 2 q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 96]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 96]][q]
Out[12]=	-12 -6 4 6 8 12 16 1 - q + q + 2 q + q + q - q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 96]][a, z]
Out[13]=	2 -4 -2 2 2 z 2 2 4 2 4 6 2 - a + a - a + 6 z - -- - 3 a z + 5 z - a z + z 4 a
In[14]:=	Kauffman[Knot[11, NonAlternating, 96]][a, z]
Out[14]=	2 2 -4 -2 2 2 z 2 z 2 z 3 2 4 z z 2 - a - a + a + --- + --- - --- - 4 a z - 2 a z - 13 z + ---- - -- - 5 3 a 4 2 a a a a 3 3 3 4 4 2 2 4 z 3 z 4 z 3 3 3 4 5 z 7 z > 8 a z - ---- - ---- + ---- + 9 a z + 6 a z + 28 z - ---- + ---- + 5 3 a 4 2 a a a a 5 5 6 6 2 4 z 3 z 5 3 5 6 z 6 z 2 6 > 16 a z + -- + ---- - a z - 5 a z - 18 z + -- - ---- - 11 a z - 5 a 4 2 a a a 7 8 9 5 z 7 3 7 8 z 2 8 z 9 > ---- - 4 a z + a z + 3 z + -- + 2 a z + -- + a z a 2 a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 96]], Vassiliev[3][Knot[11, NonAlternating, 96]]}
Out[15]=	{2, 1}
In[16]:=	Kh[Knot[11, NonAlternating, 96]][q, t]
Out[16]=	1 3 1 1 1 1 1 2 1 1 - + 3 q + q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + ---- + q 9 5 7 4 5 4 5 3 3 3 3 2 2 3 q t q t q t q t q t q t q t q t 1 2 q 3 5 3 2 5 2 7 2 7 3 7 4 > --- + --- + q t + q t + q t + q t + q t + q t + q t + q t + q t t 11 5 > q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n96

K11n95 K11n97