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

K11n32 K11n34

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X5,12,6,13 X2837 X9,17,10,16 X11,6,12,7 X13,20,14,21 X15,11,16,10 X17,1,18,22 X19,14,20,15 X21,19,22,18

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 + 12t-1 - 13 + 12t - 6t2 + t3

Conway Polynomial:

1 - 3z2 + z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{51, 2}

Jones Polynomial:

q-3 - 2q-2 + 5q-1 - 7 + 8q - 9q2 + 8q3 - 6q4 + 4q5 - q6

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

{...}

A2 (sl(3)) Invariant:

q-10 + q-8 + 2q-4 - q-2 - 3q4 + q6 - 2q8 + 2q10 + q12 + 2q16 - q18

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

a-7z3 - 2a-6z2 + 4a-6z4 + a-5z - a-5z3 + 2a-5z5 + a-5z7 + a-4 - 8a-4z2 + 10a-4z4 - 4a-4z6 + 2a-4z8 + 5a-3z - 9a-3z3 + 3a-3z5 + a-3z9 - 4a-2z2 + 5a-2z4 - 8a-2z6 + 4a-2z8 + 4a-1z - 3a-1z3 - 5a-1z5 + a-1z7 + a-1z9 - 2 + 7z2 - 5z4 - 3z6 + 2z8 + 4az3 - 6az5 + 2az7 - 2a2 + 5a2z2 - 4a2z4 + a2z6

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

{-3, -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 1133. 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 = 13                   1 j = 11                 3   j = 9               3 1   j = 7             5 3     j = 5           4 3       j = 3         4 5         j = 1       4 5           j = -1     1 3             j = -3   1 4               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, 33]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 33]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 12, 6, 13], X[2, 8, 3, 7], > X[9, 17, 10, 16], X[11, 6, 12, 7], X[13, 20, 14, 21], X[15, 11, 16, 10], > X[17, 1, 18, 22], X[19, 14, 20, 15], X[21, 19, 22, 18]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 33]]
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, 33]]
Out[5]=	DTCode[4, 8, -12, 2, -16, -6, -20, -10, -22, -14, -18]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 33]][t]
Out[6]=	-3 6 12 2 3 -13 + t - -- + -- + 12 t - 6 t + t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 33]][z]
Out[7]=	2 6 1 - 3 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 33]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 33]], KnotSignature[Knot[11, NonAlternating, 33]]}
Out[9]=	{51, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 33]][q]
Out[10]=	-3 2 5 2 3 4 5 6 -7 + q - -- + - + 8 q - 9 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, 33]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 33]][q]
Out[12]=	-10 -8 2 -2 4 6 8 10 12 16 18 q + q + -- - q - 3 q + q - 2 q + 2 q + q + 2 q - q 4 q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 33]][a, z]
Out[13]=	2 2 4 4 6 -4 2 2 z 2 z 2 2 4 z 3 z z -2 + a + 2 a - 5 z - -- + ---- + a z - 2 z - -- + ---- + -- 4 2 4 2 2 a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 33]][a, z]
Out[14]=	2 2 2 3 -4 2 z 5 z 4 z 2 2 z 8 z 4 z 2 2 z -2 + a - 2 a + -- + --- + --- + 7 z - ---- - ---- - ---- + 5 a z + -- - 5 3 a 6 4 2 7 a a a a a a 3 3 3 4 4 4 5 z 9 z 3 z 3 4 4 z 10 z 5 z 2 4 2 z > -- - ---- - ---- + 4 a z - 5 z + ---- + ----- + ---- - 4 a z + ---- + 5 3 a 6 4 2 5 a a a a a a 5 5 6 6 7 7 3 z 5 z 5 6 4 z 8 z 2 6 z z 7 > ---- - ---- - 6 a z - 3 z - ---- - ---- + a z + -- + -- + 2 a z + 3 a 4 2 5 a a a a a 8 8 9 9 8 2 z 4 z z z > 2 z + ---- + ---- + -- + -- 4 2 3 a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 33]], Vassiliev[3][Knot[11, NonAlternating, 33]]}
Out[15]=	{-3, -1}
In[16]:=	Kh[Knot[11, NonAlternating, 33]][q, t]
Out[16]=	3 1 1 1 4 1 3 4 q 3 5 q + 4 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 5 q t + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 13 5 > 4 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 K11n33

K11n32 K11n34