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K11a31 K11a33

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X14,6,15,5 X2837 X16,10,17,9 X18,11,19,12 X6,14,7,13 X22,16,1,15 X20,17,21,18 X12,19,13,20 X10,22,11,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 14 2 16 18 6 22 20 12 10

Alexander Polynomial:

3t-3 - 14t-2 + 32t-1 - 41 + 32t - 14t2 + 3t3

Conway Polynomial:

1 + 3z2 + 4z4 + 3z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{139, 2}

Jones Polynomial:

- q-2 + 4q-1 - 9 + 15q - 19q2 + 23q3 - 22q4 + 19q5 - 14q6 + 8q7 - 4q8 + q9

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

{...}

A2 (sl(3)) Invariant:

- q-6 + 2q-4 - q-2 - 2 + 4q2 - 4q4 + 3q6 + 2q8 + 5q12 - 3q14 + 3q16 - 2q18 - 4q20 + 2q22 - 2q24 + q28

HOMFLY-PT Polynomial:

a-8 + a-8z2 - 5a-6 - 7a-6z2 - 3a-6z4 + 6a-4 + 11a-4z2 + 7a-4z4 + 2a-4z6 - a-2 - a-2z2 + a-2z4 + a-2z6 - z2 - z4

Kauffman Polynomial:

a-10z2 - 2a-10z4 + a-10z6 - 2a-9z + 8a-9z3 - 10a-9z5 + 4a-9z7 + a-8 - a-8z2 + 6a-8z4 - 12a-8z6 + 6a-8z8 - 10a-7z + 31a-7z3 - 31a-7z5 + 5a-7z7 + 4a-7z9 + 5a-6 - 16a-6z2 + 35a-6z4 - 43a-6z6 + 17a-6z8 + a-6z10 - 13a-5z + 41a-5z3 - 40a-5z5 + 2a-5z7 + 9a-5z9 + 6a-4 - 19a-4z2 + 37a-4z4 - 45a-4z6 + 20a-4z8 + a-4z10 - 7a-3z + 26a-3z3 - 32a-3z5 + 9a-3z7 + 5a-3z9 + a-2 - 3a-2z2 + 5a-2z4 - 11a-2z6 + 9a-2z8 - 2a-1z + 7a-1z3 - 12a-1z5 + 8a-1z7 + 2z2 - 5z4 + 4z6 - az3 + az5

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

{3, 4}

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

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 19                       1 j = 17                     3   j = 15                   5 1   j = 13                 9 3     j = 11               10 5       j = 9             12 9         j = 7           11 10           j = 5         8 12             j = 3       7 11               j = 1     3 9                 j = -1   1 6                   j = -3   3                     j = -5 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, Alternating, 32]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 32]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[14, 6, 15, 5], X[2, 8, 3, 7], > X[16, 10, 17, 9], X[18, 11, 19, 12], X[6, 14, 7, 13], X[22, 16, 1, 15], > X[20, 17, 21, 18], X[12, 19, 13, 20], X[10, 22, 11, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 32]]
Out[4]=	GaussCode[1, -4, 2, -1, 3, -7, 4, -2, 5, -11, 6, -10, 7, -3, 8, -5, 9, -6, 10, > -9, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 32]]
Out[5]=	DTCode[4, 8, 14, 2, 16, 18, 6, 22, 20, 12, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 32]][t]
Out[6]=	3 14 32 2 3 -41 + -- - -- + -- + 32 t - 14 t + 3 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 32]][z]
Out[7]=	2 4 6 1 + 3 z + 4 z + 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 32]}
In[9]:=	{KnotDet[Knot[11, Alternating, 32]], KnotSignature[Knot[11, Alternating, 32]]}
Out[9]=	{139, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 32]][q]
Out[10]=	-2 4 2 3 4 5 6 7 8 9 -9 - q + - + 15 q - 19 q + 23 q - 22 q + 19 q - 14 q + 8 q - 4 q + q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 32]}
In[12]:=	A2Invariant[Knot[11, Alternating, 32]][q]
Out[12]=	-6 2 -2 2 4 6 8 12 14 16 -2 - q + -- - q + 4 q - 4 q + 3 q + 2 q + 5 q - 3 q + 3 q - 4 q 18 20 22 24 28 > 2 q - 4 q + 2 q - 2 q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 32]][a, z]
Out[13]=	2 2 2 2 4 4 4 -8 5 6 -2 2 z 7 z 11 z z 4 3 z 7 z z a - -- + -- - a - z + -- - ---- + ----- - -- - z - ---- + ---- + -- + 6 4 8 6 4 2 6 4 2 a a a a a a a a a 6 6 2 z z > ---- + -- 4 2 a a
In[14]:=	Kauffman[Knot[11, Alternating, 32]][a, z]
Out[14]=	2 2 2 -8 5 6 -2 2 z 10 z 13 z 7 z 2 z 2 z z 16 z a + -- + -- + a - --- - ---- - ---- - --- - --- + 2 z + --- - -- - ----- - 6 4 9 7 5 3 a 10 8 6 a a a a a a a a a 2 2 3 3 3 3 3 4 19 z 3 z 8 z 31 z 41 z 26 z 7 z 3 4 2 z > ----- - ---- + ---- + ----- + ----- + ----- + ---- - a z - 5 z - ---- + 4 2 9 7 5 3 a 10 a a a a a a a 4 4 4 4 5 5 5 5 5 6 z 35 z 37 z 5 z 10 z 31 z 40 z 32 z 12 z > ---- + ----- + ----- + ---- - ----- - ----- - ----- - ----- - ----- + 8 6 4 2 9 7 5 3 a a a a a a a a a 6 6 6 6 6 7 7 7 5 6 z 12 z 43 z 45 z 11 z 4 z 5 z 2 z > a z + 4 z + --- - ----- - ----- - ----- - ----- + ---- + ---- + ---- + 10 8 6 4 2 9 7 5 a a a a a a a a 7 7 8 8 8 8 9 9 9 10 10 9 z 8 z 6 z 17 z 20 z 9 z 4 z 9 z 5 z z z > ---- + ---- + ---- + ----- + ----- + ---- + ---- + ---- + ---- + --- + --- 3 a 8 6 4 2 7 5 3 6 4 a a a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 32]], Vassiliev[3][Knot[11, Alternating, 32]]}
Out[15]=	{3, 4}
In[16]:=	Kh[Knot[11, Alternating, 32]][q, t]
Out[16]=	3 1 3 1 6 3 q 3 5 5 2 9 q + 7 q + ----- + ----- + ---- + --- + --- + 11 q t + 8 q t + 12 q t + 5 3 3 2 2 q t t q t q t q t 7 2 7 3 9 3 9 4 11 4 11 5 > 11 q t + 10 q t + 12 q t + 9 q t + 10 q t + 5 q t + 13 5 13 6 15 6 15 7 17 7 19 8 > 9 q t + 3 q t + 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a32

K11a31 K11a33