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

K11a7 K11a9

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8394 X10,6,11,5 X16,8,17,7 X2,9,3,10 X18,11,19,12 X20,13,21,14 X6,16,7,15 X22,17,1,18 X14,19,15,20 X12,21,13,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 10 16 2 18 20 6 22 14 12

Alexander Polynomial:

- 2t-3 + 11t-2 - 27t-1 + 37 - 27t + 11t2 - 2t3

Conway Polynomial:

1 - z2 - z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11a38, K11a187, K11a249, ...}

Determinant and Signature:

{117, 0}

Jones Polynomial:

- q-7 + 3q-6 - 6q-5 + 11q-4 - 15q-3 + 18q-2 - 19q-1 + 17 - 13q + 9q2 - 4q3 + q4

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

{...}

A2 (sl(3)) Invariant:

- q-22 + q-18 - 2q-16 + 3q-14 + 2q-12 - 2q-10 + 3q-8 - 3q-6 - 2 + 4q2 - 3q4 + 2q6 + 2q8 - 2q10 + q12

HOMFLY-PT Polynomial:

a-2 + a-2z2 + a-2z4 - 2z2 - 2z4 - z6 - 2a2 - 3a2z2 - 2a2z4 - a2z6 + 3a4 + 4a4z2 + 2a4z4 - a6 - a6z2

Kauffman Polynomial:

a-4z4 - a-3z3 + 4a-3z5 - a-2 + 4a-2z2 - 9a-2z4 + 9a-2z6 - 2a-1z + 8a-1z3 - 16a-1z5 + 12a-1z7 + 3z2 - 5z4 - 9z6 + 10z8 - 4az + 21az3 - 32az5 + 7az7 + 5az9 + 2a2 - 13a2z2 + 35a2z4 - 45a2z6 + 16a2z8 + a2z10 - 5a3z + 21a3z3 - 13a3z5 - 12a3z7 + 8a3z9 + 3a4 - 20a4z2 + 46a4z4 - 39a4z6 + 9a4z8 + a4z10 - 5a5z + 14a5z3 - 5a5z5 - 6a5z7 + 3a5z9 + a6 - 8a6z2 + 16a6z4 - 12a6z6 + 3a6z8 - 2a7z + 5a7z3 - 4a7z5 + a7z7

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=0 is the signature of 118. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 9                       1 j = 7                     3   j = 5                   6 1   j = 3                 7 3     j = 1               10 6       j = -1             10 8         j = -3           8 9           j = -5         7 10             j = -7       4 8               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, 8]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 8]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[10, 6, 11, 5], X[16, 8, 17, 7], > X[2, 9, 3, 10], X[18, 11, 19, 12], X[20, 13, 21, 14], X[6, 16, 7, 15], > X[22, 17, 1, 18], X[14, 19, 15, 20], X[12, 21, 13, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 8]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -8, 4, -2, 5, -3, 6, -11, 7, -10, 8, -4, 9, -6, 10, > -7, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 8]]
Out[5]=	DTCode[4, 8, 10, 16, 2, 18, 20, 6, 22, 14, 12]
In[6]:=	alex = Alexander[Knot[11, Alternating, 8]][t]
Out[6]=	2 11 27 2 3 37 - -- + -- - -- - 27 t + 11 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 8]][z]
Out[7]=	2 4 6 1 - z - z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 8], Knot[11, Alternating, 38], > Knot[11, Alternating, 187], Knot[11, Alternating, 249]}
In[9]:=	{KnotDet[Knot[11, Alternating, 8]], KnotSignature[Knot[11, Alternating, 8]]}
Out[9]=	{117, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 8]][q]
Out[10]=	-7 3 6 11 15 18 19 2 3 4 17 - q + -- - -- + -- - -- + -- - -- - 13 q + 9 q - 4 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, 8]}
In[12]:=	A2Invariant[Knot[11, Alternating, 8]][q]
Out[12]=	-22 -18 2 3 2 2 3 3 2 4 6 -2 - q + q - --- + --- + --- - --- + -- - -- + 4 q - 3 q + 2 q + 16 14 12 10 8 6 q q q q q q 8 10 12 > 2 q - 2 q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 8]][a, z]
Out[13]=	2 4 -2 2 4 6 2 z 2 2 4 2 6 2 4 z a - 2 a + 3 a - a - 2 z + -- - 3 a z + 4 a z - a z - 2 z + -- - 2 2 a a 2 4 4 4 6 2 6 > 2 a z + 2 a z - z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 8]][a, z]
Out[14]=	-2 2 4 6 2 z 3 5 7 2 -a + 2 a + 3 a + a - --- - 4 a z - 5 a z - 5 a z - 2 a z + 3 z + a 2 3 3 4 z 2 2 4 2 6 2 z 8 z 3 3 3 > ---- - 13 a z - 20 a z - 8 a z - -- + ---- + 21 a z + 21 a z + 2 3 a a a 4 4 5 3 7 3 4 z 9 z 2 4 4 4 6 4 > 14 a z + 5 a z - 5 z + -- - ---- + 35 a z + 46 a z + 16 a z + 4 2 a a 5 5 6 4 z 16 z 5 3 5 5 5 7 5 6 9 z > ---- - ----- - 32 a z - 13 a z - 5 a z - 4 a z - 9 z + ---- - 3 a 2 a a 7 2 6 4 6 6 6 12 z 7 3 7 5 7 > 45 a z - 39 a z - 12 a z + ----- + 7 a z - 12 a z - 6 a z + a 7 7 8 2 8 4 8 6 8 9 3 9 5 9 > a z + 10 z + 16 a z + 9 a z + 3 a z + 5 a z + 8 a z + 3 a z + 2 10 4 10 > a z + a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 8]], Vassiliev[3][Knot[11, Alternating, 8]]}
Out[15]=	{-1, -1}
In[16]:=	Kh[Knot[11, Alternating, 8]][q, t]
Out[16]=	8 1 2 1 4 2 7 4 8 - + 10 q + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + q 15 7 13 6 11 6 11 5 9 5 9 4 7 4 7 3 q t q t q t q t q t q t q t q t 7 10 8 9 10 3 3 2 5 2 > ----- + ----- + ----- + ---- + --- + 6 q t + 7 q t + 3 q t + 6 q t + 5 3 5 2 3 2 3 q t q t q t q t q t 5 3 7 3 9 4 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a8

K11a7 K11a9