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

K11n91 K11n93

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X12,6,13,5 X7,20,8,21 X9,16,10,17 X2,11,3,12 X13,19,14,18 X15,8,16,9 X17,1,18,22 X19,6,20,7 X21,15,22,14

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

t-3 - 3t-2 + 3t-1 - 1 + 3t - 3t2 + t3

Conway Polynomial:

1 + 3z4 + z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{15, -2}

Jones Polynomial:

q-3 - 2q-2 + 2q-1 - 2 + 3q - 2q2 + 2q3 - q4

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

{10136, ...}

A2 (sl(3)) Invariant:

q-14 - q-8 - q-6 + 2 + q2 + q4 + q6 - q12

HOMFLY-PT Polynomial:

- a-2 - 3a-2z2 - a-2z4 + 4 + 7z2 + 5z4 + z6 - 3a2 - 4a2z2 - a2z4 + a4

Kauffman Polynomial:

- a-3z + 6a-3z3 - 5a-3z5 + a-3z7 + a-2 - 9a-2z2 + 17a-2z4 - 11a-2z6 + 2a-2z8 - a-1z + 4a-1z3 + a-1z5 - 4a-1z7 + a-1z9 + 4 - 18z2 + 28z4 - 17z6 + 3z8 + az - 3az3 + 6az5 - 5az7 + az9 + 3a2 - 10a2z2 + 11a2z4 - 6a2z6 + a2z8 + a3z - a3z3 + a4 - a4z2

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

{0, 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 1192. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 9               1 j = 7             1   j = 5           1 1   j = 3         2 1     j = 1     1 1 1       j = -1     2 2         j = -3   1 1           j = -5   1             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, 92]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 92]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[7, 20, 8, 21], > X[9, 16, 10, 17], X[2, 11, 3, 12], X[13, 19, 14, 18], X[15, 8, 16, 9], > X[17, 1, 18, 22], X[19, 6, 20, 7], X[21, 15, 22, 14]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 92]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, 10, -4, 8, -5, -2, 6, -3, -7, 11, -8, 5, -9, 7, -10, > 4, -11, 9]
In[5]:=	DTCode[Knot[11, NonAlternating, 92]]
Out[5]=	DTCode[4, 10, 12, -20, -16, 2, -18, -8, -22, -6, -14]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 92]][t]
Out[6]=	-3 3 3 2 3 -1 + t - -- + - + 3 t - 3 t + t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 92]][z]
Out[7]=	4 6 1 + 3 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 92]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 92]], KnotSignature[Knot[11, NonAlternating, 92]]}
Out[9]=	{15, -2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 92]][q]
Out[10]=	-3 2 2 2 3 4 -2 + q - -- + - + 3 q - 2 q + 2 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 136], Knot[11, NonAlternating, 92]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 92]][q]
Out[12]=	-14 -8 -6 2 4 6 12 2 + q - q - q + q + q + q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 92]][a, z]
Out[13]=	2 4 -2 2 4 2 3 z 2 2 4 z 2 4 6 4 - a - 3 a + a + 7 z - ---- - 4 a z + 5 z - -- - a z + z 2 2 a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 92]][a, z]
Out[14]=	2 -2 2 4 z z 3 2 9 z 2 2 4 2 4 + a + 3 a + a - -- - - + a z + a z - 18 z - ---- - 10 a z - a z + 3 a 2 a a 3 3 4 5 5 6 z 4 z 3 3 3 4 17 z 2 4 5 z z > ---- + ---- - 3 a z - a z + 28 z + ----- + 11 a z - ---- + -- + 3 a 2 3 a a a a 6 7 7 8 5 6 11 z 2 6 z 4 z 7 8 2 z > 6 a z - 17 z - ----- - 6 a z + -- - ---- - 5 a z + 3 z + ---- + 2 3 a 2 a a a 9 2 8 z 9 > a z + -- + a z a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 92]], Vassiliev[3][Knot[11, NonAlternating, 92]]}
Out[15]=	{0, 1}
In[16]:=	Kh[Knot[11, NonAlternating, 92]][q, t]
Out[16]=	-3 2 1 1 1 2 t 2 3 2 3 3 q + - + q + ----- + ---- + ---- + --- + q t + q t + 2 q t + q t + q 7 2 5 3 q q t q t q t 5 3 5 4 7 4 9 5 > q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n92

K11n91 K11n93