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

K11n31 K11n33

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X5,12,6,13 X2837 X16,9,17,10 X11,6,12,7 X20,14,21,13 X10,15,11,16 X22,17,1,18 X14,20,15,19 X18,21,19,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- t-3 + 6t-2 - 16t-1 + 23 - 16t + 6t2 - t3

Conway Polynomial:

1 - z2 - z6

Other knots with the same Alexander/Conway Polynomial:

{934, K11n119, ...}

Determinant and Signature:

{69, 0}

Jones Polynomial:

2q-4 - 5q-3 + 8q-2 - 11q-1 + 12 - 11q + 10q2 - 6q3 + 3q4 - q5

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

{...}

A2 (sl(3)) Invariant:

q-14 + 2q-12 - 2q-10 + q-8 - q-6 - 3q-4 + 2q-2 - 2 + 3q2 + q6 + 3q8 - 2q10 + q12 - q16

HOMFLY-PT Polynomial:

- a-4 - a-4z2 + 3a-2 + 4a-2z2 + 2a-2z4 - 1 - 4z2 - 3z4 - z6 - a2 + a2z4 + a4

Kauffman Polynomial:

a-5z - 2a-5z3 + a-5z5 - a-4 + 4a-4z2 - 6a-4z4 + 3a-4z6 + 2a-3z - a-3z3 - 5a-3z5 + 4a-3z7 - 3a-2 + 8a-2z2 - 10a-2z4 + a-2z6 + 3a-2z8 + 3a-1z - a-1z3 - 9a-1z5 + 6a-1z7 + a-1z9 - 1 + 2z2 - 3z4 - 3z6 + 5z8 + 5az - 8az3 + 3az7 + az9 + a2 - 6a2z2 + 4a2z4 - a2z6 + 2a2z8 + 3a3z - 6a3z3 + 3a3z5 + a3z7 + a4 - 4a4z2 + 3a4z4

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

{-1, 2}

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 1132. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj 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                 2   j = 7               4 1   j = 5             6 2     j = 3           5 4       j = 1         7 6         j = -1       5 6           j = -3     3 6             j = -5   2 5               j = -7   3                 j = -9 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, 32]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 32]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 12, 6, 13], X[2, 8, 3, 7], > X[16, 9, 17, 10], X[11, 6, 12, 7], X[20, 14, 21, 13], X[10, 15, 11, 16], > X[22, 17, 1, 18], X[14, 20, 15, 19], X[18, 21, 19, 22]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 32]]
Out[4]=	GaussCode[1, -4, 2, -1, -3, 6, 4, -2, 5, -8, -6, 3, 7, -10, 8, -5, 9, -11, 10, > -7, 11, -9]
In[5]:=	DTCode[Knot[11, NonAlternating, 32]]
Out[5]=	DTCode[4, 8, -12, 2, 16, -6, 20, 10, 22, 14, 18]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 32]][t]
Out[6]=	-3 6 16 2 3 23 - t + -- - -- - 16 t + 6 t - t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 32]][z]
Out[7]=	2 6 1 - z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 34], Knot[11, NonAlternating, 32], Knot[11, NonAlternating, 119]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 32]], KnotSignature[Knot[11, NonAlternating, 32]]}
Out[9]=	{69, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 32]][q]
Out[10]=	2 5 8 11 2 3 4 5 12 + -- - -- + -- - -- - 11 q + 10 q - 6 q + 3 q - q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 32]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 32]][q]
Out[12]=	-14 2 2 -8 -6 3 2 2 6 8 10 12 -2 + q + --- - --- + q - q - -- + -- + 3 q + q + 3 q - 2 q + q - 12 10 4 2 q q q q 16 > q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 32]][a, z]
Out[13]=	2 2 4 -4 3 2 4 2 z 4 z 4 2 z 2 4 6 -1 - a + -- - a + a - 4 z - -- + ---- - 3 z + ---- + a z - z 2 4 2 2 a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 32]][a, z]
Out[14]=	2 -4 3 2 4 z 2 z 3 z 3 2 4 z -1 - a - -- + a + a + -- + --- + --- + 5 a z + 3 a z + 2 z + ---- + 2 5 3 a 4 a a a a 2 3 3 3 8 z 2 2 4 2 2 z z z 3 3 3 4 > ---- - 6 a z - 4 a z - ---- - -- - -- - 8 a z - 6 a z - 3 z - 2 5 3 a a a a 4 4 5 5 5 6 z 10 z 2 4 4 4 z 5 z 9 z 3 5 6 > ---- - ----- + 4 a z + 3 a z + -- - ---- - ---- + 3 a z - 3 z + 4 2 5 3 a a a a a 6 6 7 7 8 3 z z 2 6 4 z 6 z 7 3 7 8 3 z 2 8 > ---- + -- - a z + ---- + ---- + 3 a z + a z + 5 z + ---- + 2 a z + 4 2 3 a 2 a a a a 9 z 9 > -- + a z a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 32]], Vassiliev[3][Knot[11, NonAlternating, 32]]}
Out[15]=	{-1, 2}
In[16]:=	Kh[Knot[11, NonAlternating, 32]][q, t]
Out[16]=	6 2 3 2 5 3 6 5 3 - + 7 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 6 q t + 5 q t + q 9 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 2 5 2 5 3 7 3 7 4 9 4 11 5 > 4 q t + 6 q t + 2 q t + 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n32

K11n31 K11n33