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

K11n94 K11n96

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,4,11,3 X14,6,15,5 X7,13,8,12 X2,10,3,9 X11,21,12,20 X18,14,19,13 X15,22,16,1 X6,18,7,17 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 + 5t-2 - 7t-1 + 7 - 7t + 5t2 - t3

Conway Polynomial:

1 + 4z2 - z4 - z6

Other knots with the same Alexander/Conway Polynomial:

{911, ...}

Determinant and Signature:

{33, 4}

Jones Polynomial:

2q2 - 3q3 + 5q4 - 6q5 + 6q6 - 5q7 + 4q8 - 2q9

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

{...}

A2 (sl(3)) Invariant:

2q6 + 2q10 + q16 - 2q18 + q20 - q22 + q24 + q26 - q28 - q32

HOMFLY-PT Polynomial:

- a-10 + 2a-8 + 3a-8z2 + a-8z4 - 3a-6 - 5a-6z2 - 4a-6z4 - a-6z6 + 3a-4 + 6a-4z2 + 2a-4z4

Kauffman Polynomial:

- 3a-11z + 3a-11z3 + a-10 - 4a-10z2 + 2a-10z4 + a-10z6 - 3a-9z + 7a-9z3 - 4a-9z5 + 2a-9z7 + 2a-8 - 6a-8z2 + 5a-8z4 - a-8z6 + a-8z8 - a-7z + 4a-7z3 - 5a-7z5 + 3a-7z7 + 3a-6 - 9a-6z2 + 6a-6z4 - 2a-6z6 + a-6z8 - a-5z - a-5z5 + a-5z7 + 3a-4 - 7a-4z2 + 3a-4z4

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

{4, 9}

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=4 is the signature of 1195. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 19               2 j = 17             2   j = 15           3 2   j = 13         3 2     j = 11       3 3       j = 9     2 3         j = 7   1 3           j = 5 1 2             j = 3 2

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, 95]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 95]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[14, 6, 15, 5], X[7, 13, 8, 12], > X[2, 10, 3, 9], X[11, 21, 12, 20], X[18, 14, 19, 13], X[15, 22, 16, 1], > X[6, 18, 7, 17], X[19, 9, 20, 8], X[21, 16, 22, 17]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 95]]
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, 95]]
Out[5]=	DTCode[4, 10, 14, -12, 2, -20, 18, -22, 6, -8, -16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 95]][t]
Out[6]=	-3 5 7 2 3 7 - t + -- - - - 7 t + 5 t - t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 95]][z]
Out[7]=	2 4 6 1 + 4 z - z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 11], Knot[11, NonAlternating, 95]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 95]], KnotSignature[Knot[11, NonAlternating, 95]]}
Out[9]=	{33, 4}
In[10]:=	J=Jones[Knot[11, NonAlternating, 95]][q]
Out[10]=	2 3 4 5 6 7 8 9 2 q - 3 q + 5 q - 6 q + 6 q - 5 q + 4 q - 2 q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 95]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 95]][q]
Out[12]=	6 10 16 18 20 22 24 26 28 32 2 q + 2 q + q - 2 q + q - q + q + q - q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 95]][a, z]
Out[13]=	2 2 2 4 4 4 6 -10 2 3 3 3 z 5 z 6 z z 4 z 2 z z -a + -- - -- + -- + ---- - ---- + ---- + -- - ---- + ---- - -- 8 6 4 8 6 4 8 6 4 6 a a a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 95]][a, z]
Out[14]=	2 2 2 2 3 -10 2 3 3 3 z 3 z z z 4 z 6 z 9 z 7 z 3 z a + -- + -- + -- - --- - --- - -- - -- - ---- - ---- - ---- - ---- + ---- + 8 6 4 11 9 7 5 10 8 6 4 11 a a a a a a a a a a a a 3 3 4 4 4 4 5 5 5 6 6 7 z 4 z 2 z 5 z 6 z 3 z 4 z 5 z z z z > ---- + ---- + ---- + ---- + ---- + ---- - ---- - ---- - -- + --- - -- - 9 7 10 8 6 4 9 7 5 10 8 a a a a a a a a a a a 6 7 7 7 8 8 2 z 2 z 3 z z z z > ---- + ---- + ---- + -- + -- + -- 6 9 7 5 8 6 a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 95]], Vassiliev[3][Knot[11, NonAlternating, 95]]}
Out[15]=	{4, 9}
In[16]:=	Kh[Knot[11, NonAlternating, 95]][q, t]
Out[16]=	3 5 5 7 7 2 9 2 9 3 11 3 11 4 2 q + q + 2 q t + q t + 3 q t + 2 q t + 3 q t + 3 q t + 3 q t + 13 4 13 5 15 5 15 6 17 6 19 7 > 3 q t + 2 q t + 3 q t + 2 q t + 2 q t + 2 q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n95

K11n94 K11n96