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

K11a108 K11a110

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 14 16 2 20 22 18 8 6 12

Alexander Polynomial:

t-4 - 5t-3 + 14t-2 - 24t-1 + 29 - 24t + 14t2 - 5t3 + t4

Conway Polynomial:

1 + 3z2 + 4z4 + 3z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{K11a44, K11a47, ...}

Determinant and Signature:

{117, 0}

Jones Polynomial:

- q-5 + 3q-4 - 7q-3 + 12q-2 - 16q-1 + 19 - 18q + 17q2 - 12q3 + 7q4 - 4q5 + q6

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

{...}

A2 (sl(3)) Invariant:

- q-14 + q-12 - 3q-10 + q-8 + q-6 - 2q-4 + 5q-2 - 2 + 4q2 + q4 + 3q8 - 4q10 - q14 - q16 + q18

HOMFLY-PT Polynomial:

2a-4z2 + a-4z4 - 3a-2 - 10a-2z2 - 8a-2z4 - 2a-2z6 + 7 + 17z2 + 15z4 + 6z6 + z8 - 3a2 - 6a2z2 - 4a2z4 - a2z6

Kauffman Polynomial:

- 2a-6z4 + a-6z6 + 7a-5z3 - 11a-5z5 + 4a-5z7 - 5a-4z2 + 13a-4z4 - 16a-4z6 + 6a-4z8 - 4a-3z + 19a-3z3 - 18a-3z5 - a-3z7 + 4a-3z9 + 3a-2 - 22a-2z2 + 46a-2z4 - 42a-2z6 + 13a-2z8 + a-2z10 - 8a-1z + 20a-1z3 - 10a-1z5 - 9a-1z7 + 8a-1z9 + 7 - 27z2 + 45z4 - 37z6 + 13z8 + z10 - 6az + 13az3 - 11az5 + az7 + 4az9 + 3a2 - 8a2z2 + 9a2z4 - 9a2z6 + 6a2z8 - a3z + 3a3z3 - 7a3z5 + 5a3z7 + 2a4z2 - 5a4z4 + 3a4z6 + a5z - 2a5z3 + a5z5

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

{3, 0}

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 11109. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 13                       1 j = 11                     3   j = 9                   4 1   j = 7                 8 3     j = 5               9 4       j = 3             9 8         j = 1           10 9           j = -1         7 10             j = -3       5 9               j = -5     2 7                 j = -7   1 5                   j = -9   2                     j = -11 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, 109]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 109]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[14, 5, 15, 6], X[16, 8, 17, 7], > X[2, 10, 3, 9], X[20, 11, 21, 12], X[22, 13, 1, 14], X[18, 16, 19, 15], > X[8, 18, 9, 17], X[6, 19, 7, 20], X[12, 21, 13, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 109]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -10, 4, -9, 5, -2, 6, -11, 7, -3, 8, -4, 9, -8, 10, > -6, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 109]]
Out[5]=	DTCode[4, 10, 14, 16, 2, 20, 22, 18, 8, 6, 12]
In[6]:=	alex = Alexander[Knot[11, Alternating, 109]][t]
Out[6]=	-4 5 14 24 2 3 4 29 + t - -- + -- - -- - 24 t + 14 t - 5 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 109]][z]
Out[7]=	2 4 6 8 1 + 3 z + 4 z + 3 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 44], Knot[11, Alternating, 47], > Knot[11, Alternating, 109]}
In[9]:=	{KnotDet[Knot[11, Alternating, 109]], KnotSignature[Knot[11, Alternating, 109]]}
Out[9]=	{117, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 109]][q]
Out[10]=	-5 3 7 12 16 2 3 4 5 6 19 - q + -- - -- + -- - -- - 18 q + 17 q - 12 q + 7 q - 4 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, Alternating, 109]}
In[12]:=	A2Invariant[Knot[11, Alternating, 109]][q]
Out[12]=	-14 -12 3 -8 -6 2 5 2 4 8 10 14 -2 - q + q - --- + q + q - -- + -- + 4 q + q + 3 q - 4 q - q - 10 4 2 q q q 16 18 > q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 109]][a, z]
Out[13]=	2 2 4 4 3 2 2 2 z 10 z 2 2 4 z 8 z 2 4 7 - -- - 3 a + 17 z + ---- - ----- - 6 a z + 15 z + -- - ---- - 4 a z + 2 4 2 4 2 a a a a a 6 6 2 z 2 6 8 > 6 z - ---- - a z + z 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 109]][a, z]
Out[14]=	2 2 3 2 4 z 8 z 3 5 2 5 z 22 z 7 + -- + 3 a - --- - --- - 6 a z - a z + a z - 27 z - ---- - ----- - 2 3 a 4 2 a a a a 3 3 3 2 2 4 2 7 z 19 z 20 z 3 3 3 5 3 > 8 a z + 2 a z + ---- + ----- + ----- + 13 a z + 3 a z - 2 a z + 5 3 a a a 4 4 4 5 5 5 4 2 z 13 z 46 z 2 4 4 4 11 z 18 z 10 z > 45 z - ---- + ----- + ----- + 9 a z - 5 a z - ----- - ----- - ----- - 6 4 2 5 3 a a a a a a 6 6 6 5 3 5 5 5 6 z 16 z 42 z 2 6 > 11 a z - 7 a z + a z - 37 z + -- - ----- - ----- - 9 a z + 6 4 2 a a a 7 7 7 8 8 4 6 4 z z 9 z 7 3 7 8 6 z 13 z > 3 a z + ---- - -- - ---- + a z + 5 a z + 13 z + ---- + ----- + 5 3 a 4 2 a a a a 9 9 10 2 8 4 z 8 z 9 10 z > 6 a z + ---- + ---- + 4 a z + z + --- 3 a 2 a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 109]], Vassiliev[3][Knot[11, Alternating, 109]]}
Out[15]=	{3, 0}
In[16]:=	Kh[Knot[11, Alternating, 109]][q, t]
Out[16]=	10 1 2 1 5 2 7 5 9 -- + 10 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t 7 3 3 2 5 2 5 3 7 3 7 4 > --- + 9 q t + 9 q t + 8 q t + 9 q t + 4 q t + 8 q t + 3 q t + q t 9 4 9 5 11 5 13 6 > 4 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a109

K11a108 K11a110