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

K11a33 K11a35

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 14 2 16 18 20 6 22 12 10

Alexander Polynomial:

- t-4 + 5t-3 - 14t-2 + 25t-1 - 29 + 25t - 14t2 + 5t3 - t4

Conway Polynomial:

1 - 2z2 - 4z4 - 3z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{K11a158, ...}

Determinant and Signature:

{119, 2}

Jones Polynomial:

q-3 - 3q-2 + 7q-1 - 11 + 16q - 19q2 + 19q3 - 17q4 + 13q5 - 8q6 + 4q7 - q8

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

{K11a89, ...}

A2 (sl(3)) Invariant:

q-8 - q-6 + 3q-4 + 3q2 - 5q4 + 2q6 - 3q8 + 2q12 - 2q14 + 4q16 - q18 + q22 - q24

HOMFLY-PT Polynomial:

- a-6 - 2a-6z2 - a-6z4 + 5a-4 + 11a-4z2 + 8a-4z4 + 2a-4z6 - 7a-2 - 17a-2z2 - 15a-2z4 - 6a-2z6 - a-2z8 + 4 + 6z2 + 4z4 + z6

Kauffman Polynomial:

- a-9z3 + a-9z5 + 2a-8z2 - 6a-8z4 + 4a-8z6 - 2a-7z + 5a-7z3 - 11a-7z5 + 7a-7z7 + a-6 + a-6z2 - 2a-6z4 - 7a-6z6 + 7a-6z8 - 7a-5z + 23a-5z3 - 25a-5z5 + 6a-5z7 + 4a-5z9 + 5a-4 - 20a-4z2 + 39a-4z4 - 37a-4z6 + 14a-4z8 + a-4z10 - 7a-3z + 19a-3z3 - 9a-3z5 - 9a-3z7 + 8a-3z9 + 7a-2 - 32a-2z2 + 53a-2z4 - 41a-2z6 + 12a-2z8 + a-2z10 - 3a-1z + 7a-1z3 - 4a-1z5 - 5a-1z7 + 4a-1z9 + 4 - 11z2 + 15z4 - 14z6 + 5z8 - az + 5az3 - 8az5 + 3az7 + 2a2z2 - 3a2z4 + a2z6

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

{-2, -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 1134. 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 r = 6 r = 7 j = 17                       1 j = 15                     3   j = 13                   5 1   j = 11                 8 3     j = 9               9 5       j = 7             10 8         j = 5           9 9           j = 3         7 10             j = 1       5 10               j = -1     2 6                 j = -3   1 5                   j = -5   2                     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, Alternating, 34]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 34]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[14, 5, 15, 6], X[2, 8, 3, 7], > X[16, 9, 17, 10], X[18, 12, 19, 11], X[20, 14, 21, 13], X[6, 15, 7, 16], > X[22, 17, 1, 18], X[12, 20, 13, 19], X[10, 22, 11, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 34]]
Out[4]=	GaussCode[1, -4, 2, -1, 3, -8, 4, -2, 5, -11, 6, -10, 7, -3, 8, -5, 9, -6, 10, > -7, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 34]]
Out[5]=	DTCode[4, 8, 14, 2, 16, 18, 20, 6, 22, 12, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 34]][t]
Out[6]=	-4 5 14 25 2 3 4 -29 - t + -- - -- + -- + 25 t - 14 t + 5 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 34]][z]
Out[7]=	2 4 6 8 1 - 2 z - 4 z - 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 34], Knot[11, Alternating, 158]}
In[9]:=	{KnotDet[Knot[11, Alternating, 34]], KnotSignature[Knot[11, Alternating, 34]]}
Out[9]=	{119, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 34]][q]
Out[10]=	-3 3 7 2 3 4 5 6 7 8 -11 + q - -- + - + 16 q - 19 q + 19 q - 17 q + 13 q - 8 q + 4 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 34], Knot[11, Alternating, 89]}
In[12]:=	A2Invariant[Knot[11, Alternating, 34]][q]
Out[12]=	-8 -6 3 2 4 6 8 12 14 16 18 q - q + -- + 3 q - 5 q + 2 q - 3 q + 2 q - 2 q + 4 q - q + 4 q 22 24 > q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 34]][a, z]
Out[13]=	2 2 2 4 4 4 -6 5 7 2 2 z 11 z 17 z 4 z 8 z 15 z 4 - a + -- - -- + 6 z - ---- + ----- - ----- + 4 z - -- + ---- - ----- + 4 2 6 4 2 6 4 2 a a a a a a a a 6 6 8 6 2 z 6 z z > z + ---- - ---- - -- 4 2 2 a a a
In[14]:=	Kauffman[Knot[11, Alternating, 34]][a, z]
Out[14]=	2 2 2 -6 5 7 2 z 7 z 7 z 3 z 2 2 z z 20 z 4 + a + -- + -- - --- - --- - --- - --- - a z - 11 z + ---- + -- - ----- - 4 2 7 5 3 a 8 6 4 a a a a a a a a 2 3 3 3 3 3 32 z 2 2 z 5 z 23 z 19 z 7 z 3 4 > ----- + 2 a z - -- + ---- + ----- + ----- + ---- + 5 a z + 15 z - 2 9 7 5 3 a a a a a a 4 4 4 4 5 5 5 5 5 6 z 2 z 39 z 53 z 2 4 z 11 z 25 z 9 z 4 z > ---- - ---- + ----- + ----- - 3 a z + -- - ----- - ----- - ---- - ---- - 8 6 4 2 9 7 5 3 a a a a a a a a a 6 6 6 6 7 7 7 5 6 4 z 7 z 37 z 41 z 2 6 7 z 6 z 9 z > 8 a z - 14 z + ---- - ---- - ----- - ----- + a z + ---- + ---- - ---- - 8 6 4 2 7 5 3 a a a a a a a 7 8 8 8 9 9 9 10 10 5 z 7 8 7 z 14 z 12 z 4 z 8 z 4 z z z > ---- + 3 a z + 5 z + ---- + ----- + ----- + ---- + ---- + ---- + --- + --- a 6 4 2 5 3 a 4 2 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 34]], Vassiliev[3][Knot[11, Alternating, 34]]}
Out[15]=	{-2, -1}
In[16]:=	Kh[Knot[11, Alternating, 34]][q, t]
Out[16]=	3 1 2 1 5 2 6 5 q 3 10 q + 7 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 10 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 > 9 q t + 9 q t + 10 q t + 8 q t + 9 q t + 5 q t + 8 q t + 11 5 13 5 13 6 15 6 17 7 > 3 q t + 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a34

K11a33 K11a35