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

K11a113 K11a115

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,4,11,3 X14,6,15,5 X18,7,19,8 X2,10,3,9 X16,12,17,11 X6,14,7,13 X22,16,1,15 X20,17,21,18 X8,19,9,20 X12,22,13,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 14 18 2 16 6 22 20 8 12

Alexander Polynomial:

3t-3 - 15t-2 + 35t-1 - 45 + 35t - 15t2 + 3t3

Conway Polynomial:

1 + 2z2 + 3z4 + 3z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{151, 2}

Jones Polynomial:

- q-2 + 4q-1 - 9 + 16q - 21q2 + 25q3 - 24q4 + 21q5 - 16q6 + 9q7 - 4q8 + q9

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

{...}

A2 (sl(3)) Invariant:

- q-6 + 2q-4 - q-2 - 2 + 5q2 - 4q4 + 3q6 + 2q8 - q10 + 5q12 - 4q14 + 3q16 - 2q18 - 4q20 + 3q22 - 2q24 + q28

HOMFLY-PT Polynomial:

a-8 + a-8z2 - 4a-6 - 6a-6z2 - 3a-6z4 + 4a-4 + 8a-4z2 + 6a-4z4 + 2a-4z6 + a-2z4 + a-2z6 - z2 - z4

Kauffman Polynomial:

a-10z2 - 2a-10z4 + a-10z6 - 2a-9z + 7a-9z3 - 9a-9z5 + 4a-9z7 + a-8 - 2a-8z2 + 8a-8z4 - 14a-8z6 + 7a-8z8 - 7a-7z + 20a-7z3 - 19a-7z5 - 2a-7z7 + 6a-7z9 + 4a-6 - 15a-6z2 + 35a-6z4 - 44a-6z6 + 16a-6z8 + 2a-6z10 - 7a-5z + 22a-5z3 - 18a-5z5 - 12a-5z7 + 13a-5z9 + 4a-4 - 16a-4z2 + 37a-4z4 - 46a-4z6 + 19a-4z8 + 2a-4z10 - 3a-3z + 16a-3z3 - 20a-3z5 + 2a-3z7 + 7a-3z9 - 2a-2z2 + 7a-2z4 - 13a-2z6 + 10a-2z8 - a-1z + 6a-1z3 - 11a-1z5 + 8a-1z7 + 2z2 - 5z4 + 4z6 - az3 + az5

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

{2, 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=2 is the signature of 11114. 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                   6 1   j = 13                 10 3     j = 11               11 6       j = 9             13 10         j = 7           12 11           j = 5         9 13             j = 3       7 12               j = 1     3 10                 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, 114]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 114]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[14, 6, 15, 5], X[18, 7, 19, 8], > X[2, 10, 3, 9], X[16, 12, 17, 11], X[6, 14, 7, 13], X[22, 16, 1, 15], > X[20, 17, 21, 18], X[8, 19, 9, 20], X[12, 22, 13, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 114]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -7, 4, -10, 5, -2, 6, -11, 7, -3, 8, -6, 9, -4, 10, > -9, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 114]]
Out[5]=	DTCode[4, 10, 14, 18, 2, 16, 6, 22, 20, 8, 12]
In[6]:=	alex = Alexander[Knot[11, Alternating, 114]][t]
Out[6]=	3 15 35 2 3 -45 + -- - -- + -- + 35 t - 15 t + 3 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 114]][z]
Out[7]=	2 4 6 1 + 2 z + 3 z + 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 114]}
In[9]:=	{KnotDet[Knot[11, Alternating, 114]], KnotSignature[Knot[11, Alternating, 114]]}
Out[9]=	{151, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 114]][q]
Out[10]=	-2 4 2 3 4 5 6 7 8 9 -9 - q + - + 16 q - 21 q + 25 q - 24 q + 21 q - 16 q + 9 q - 4 q + q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 114]}
In[12]:=	A2Invariant[Knot[11, Alternating, 114]][q]
Out[12]=	-6 2 -2 2 4 6 8 10 12 14 16 -2 - q + -- - q + 5 q - 4 q + 3 q + 2 q - q + 5 q - 4 q + 3 q - 4 q 18 20 22 24 28 > 2 q - 4 q + 3 q - 2 q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 114]][a, z]
Out[13]=	2 2 2 4 4 4 6 6 -8 4 4 2 z 6 z 8 z 4 3 z 6 z z 2 z z a - -- + -- - z + -- - ---- + ---- - z - ---- + ---- + -- + ---- + -- 6 4 8 6 4 6 4 2 4 2 a a a a a a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 114]][a, z]
Out[14]=	2 2 2 2 -8 4 4 2 z 7 z 7 z 3 z z 2 z 2 z 15 z 16 z a + -- + -- - --- - --- - --- - --- - - + 2 z + --- - ---- - ----- - ----- - 6 4 9 7 5 3 a 10 8 6 4 a a a a a a a a a a 2 3 3 3 3 3 4 4 2 z 7 z 20 z 22 z 16 z 6 z 3 4 2 z 8 z > ---- + ---- + ----- + ----- + ----- + ---- - a z - 5 z - ---- + ---- + 2 9 7 5 3 a 10 8 a a a a a a a 4 4 4 5 5 5 5 5 35 z 37 z 7 z 9 z 19 z 18 z 20 z 11 z 5 6 > ----- + ----- + ---- - ---- - ----- - ----- - ----- - ----- + a z + 4 z + 6 4 2 9 7 5 3 a a a a a a a a 6 6 6 6 6 7 7 7 7 7 z 14 z 44 z 46 z 13 z 4 z 2 z 12 z 2 z 8 z > --- - ----- - ----- - ----- - ----- + ---- - ---- - ----- + ---- + ---- + 10 8 6 4 2 9 7 5 3 a a a a a a a a a a 8 8 8 8 9 9 9 10 10 7 z 16 z 19 z 10 z 6 z 13 z 7 z 2 z 2 z > ---- + ----- + ----- + ----- + ---- + ----- + ---- + ----- + ----- 8 6 4 2 7 5 3 6 4 a a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 114]], Vassiliev[3][Knot[11, Alternating, 114]]}
Out[15]=	{2, 2}
In[16]:=	Kh[Knot[11, Alternating, 114]][q, t]
Out[16]=	3 1 3 1 6 3 q 3 5 5 2 10 q + 7 q + ----- + ----- + ---- + --- + --- + 12 q t + 9 q t + 13 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 > 12 q t + 11 q t + 13 q t + 10 q t + 11 q t + 6 q t + 13 5 13 6 15 6 15 7 17 7 19 8 > 10 q t + 3 q t + 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a114

K11a113 K11a115