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

K11a107 K11a109

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 X18,15,19,16 X6,18,7,17 X8,19,9,20 X12,21,13,22

Gauss Code:

{1, -5, 2, -1, 3, -9, 4, -10, 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 6 8 12

Alexander Polynomial:

- t-4 + 5t-3 - 12t-2 + 20t-1 - 23 + 20t - 12t2 + 5t3 - t4

Conway Polynomial:

1 + z2 - 2z4 - 3z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{K11a57, K11a139, K11a231, ...}

Determinant and Signature:

{99, -2}

Jones Polynomial:

q-7 - 3q-6 + 6q-5 - 11q-4 + 14q-3 - 15q-2 + 16q-1 - 13 + 10q - 6q2 + 3q3 - q4

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

{...}

A2 (sl(3)) Invariant:

q-20 - q-18 + 2q-16 - 2q-14 - 2q-12 + q-10 - 3q-8 + 4q-6 - q-4 + 2q-2 + 2 - q2 + 3q4 - q6 - q12

HOMFLY-PT Polynomial:

- 2a-2 - 3a-2z2 - a-2z4 + 6 + 13z2 + 9z4 + 2z6 - 4a2 - 14a2z2 - 14a2z4 - 6a2z6 - a2z8 + a4 + 5a4z2 + 4a4z4 + a4z6

Kauffman Polynomial:

- 2a-3z + 5a-3z3 - 4a-3z5 + a-3z7 + 2a-2 - 7a-2z2 + 15a-2z4 - 12a-2z6 + 3a-2z8 - 5a-1z + 12a-1z3 - 3a-1z5 - 7a-1z7 + 3a-1z9 + 6 - 23z2 + 42z4 - 34z6 + 7z8 + z10 - 6az + 14az3 - 4az5 - 14az7 + 7az9 + 4a2 - 23a2z2 + 41a2z4 - 38a2z6 + 11a2z8 + a2z10 - 3a3z + 15a3z3 - 17a3z5 + a3z7 + 4a3z9 + a4 - 5a4z2 + 9a4z4 - 11a4z6 + 7a4z8 + a5z + 5a5z3 - 9a5z5 + 7a5z7 + a6z2 - 4a6z4 + 5a6z6 + a7z - 3a7z3 + 3a7z5 - a8z2 + a8z4

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

{1, 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 11108. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 9                       1 j = 7                     2   j = 5                   4 1   j = 3                 6 2     j = 1               7 4       j = -1             9 6         j = -3           7 8           j = -5         7 8             j = -7       4 7               j = -9     2 7                 j = -11   1 4                   j = -13   2                     j = -15 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, 108]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 108]]
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, 15, 19, 16], > X[6, 18, 7, 17], X[8, 19, 9, 20], X[12, 21, 13, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 108]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -9, 4, -10, 5, -2, 6, -11, 7, -3, 8, -4, 9, -8, 10, > -6, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 108]]
Out[5]=	DTCode[4, 10, 14, 16, 2, 20, 22, 18, 6, 8, 12]
In[6]:=	alex = Alexander[Knot[11, Alternating, 108]][t]
Out[6]=	-4 5 12 20 2 3 4 -23 - t + -- - -- + -- + 20 t - 12 t + 5 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 108]][z]
Out[7]=	2 4 6 8 1 + z - 2 z - 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 57], Knot[11, Alternating, 108], > Knot[11, Alternating, 139], Knot[11, Alternating, 231]}
In[9]:=	{KnotDet[Knot[11, Alternating, 108]], KnotSignature[Knot[11, Alternating, 108]]}
Out[9]=	{99, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 108]][q]
Out[10]=	-7 3 6 11 14 15 16 2 3 4 -13 + q - -- + -- - -- + -- - -- + -- + 10 q - 6 q + 3 q - q 6 5 4 3 2 q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 108]}
In[12]:=	A2Invariant[Knot[11, Alternating, 108]][q]
Out[12]=	-20 -18 2 2 2 -10 3 4 -4 2 2 4 2 + q - q + --- - --- - --- + q - -- + -- - q + -- - q + 3 q - 16 14 12 8 6 2 q q q q q q 6 12 > q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 108]][a, z]
Out[13]=	2 4 2 2 4 2 3 z 2 2 4 2 4 z 2 4 6 - -- - 4 a + a + 13 z - ---- - 14 a z + 5 a z + 9 z - -- - 14 a z + 2 2 2 a a a 4 4 6 2 6 4 6 2 8 > 4 a z + 2 z - 6 a z + a z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 108]][a, z]
Out[14]=	2 2 2 4 2 z 5 z 3 5 7 2 7 z 6 + -- + 4 a + a - --- - --- - 6 a z - 3 a z + a z + a z - 23 z - ---- - 2 3 a 2 a a a 3 3 2 2 4 2 6 2 8 2 5 z 12 z 3 3 3 > 23 a z - 5 a z + a z - a z + ---- + ----- + 14 a z + 15 a z + 3 a a 4 5 3 7 3 4 15 z 2 4 4 4 6 4 8 4 > 5 a z - 3 a z + 42 z + ----- + 41 a z + 9 a z - 4 a z + a z - 2 a 5 5 6 4 z 3 z 5 3 5 5 5 7 5 6 12 z > ---- - ---- - 4 a z - 17 a z - 9 a z + 3 a z - 34 z - ----- - 3 a 2 a a 7 7 2 6 4 6 6 6 z 7 z 7 3 7 5 7 > 38 a z - 11 a z + 5 a z + -- - ---- - 14 a z + a z + 7 a z + 3 a a 8 9 8 3 z 2 8 4 8 3 z 9 3 9 10 2 10 > 7 z + ---- + 11 a z + 7 a z + ---- + 7 a z + 4 a z + z + a z 2 a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 108]], Vassiliev[3][Knot[11, Alternating, 108]]}
Out[15]=	{1, 1}
In[16]:=	Kh[Knot[11, Alternating, 108]][q, t]
Out[16]=	8 9 1 2 1 4 2 7 4 7 -- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 3 q 15 6 13 5 11 5 11 4 9 4 9 3 7 3 7 2 q q t q t q t q t q t q t q t q t 7 8 7 6 t 2 3 2 3 3 5 3 > ----- + ---- + ---- + --- + 7 q t + 4 q t + 6 q t + 2 q t + 4 q t + 5 2 5 3 q q t q t q t 5 4 7 4 9 5 > q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a108

K11a107 K11a109