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

K11a168 K11a170

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 18 16 14 2 20 22 6 12 8

Alexander Polynomial:

- 2t-3 + 12t-2 - 28t-1 + 37 - 28t + 12t2 - 2t3

Conway Polynomial:

1 + 2z2 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11a36, ...}

Determinant and Signature:

{121, 0}

Jones Polynomial:

- q-7 + 3q-6 - 7q-5 + 12q-4 - 16q-3 + 19q-2 - 19q-1 + 18 - 13q + 8q2 - 4q3 + q4

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

{...}

A2 (sl(3)) Invariant:

- q-22 + q-18 - 3q-16 + 2q-14 + q-12 - 2q-10 + 4q-8 - 2q-6 + 2q-4 + q-2 - 1 + 4q2 - 4q4 + q6 + q8 - 2q10 + q12

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

a-4z4 - 2a-3z3 + 4a-3z5 + 2a-2z2 - 7a-2z4 + 8a-2z6 + 4a-1z3 - 13a-1z5 + 11a-1z7 + 2z2 + 3z4 - 15z6 + 11z8 + 8az3 - 13az5 - 4az7 + 7az9 - a2 - 5a2z2 + 29a2z4 - 40a2z6 + 12a2z8 + 2a2z10 + a3z3 + 13a3z5 - 27a3z7 + 11a3z9 + a4 - 11a4z2 + 31a4z4 - 28a4z6 + 4a4z8 + 2a4z10 - 2a5z + 4a5z3 + 5a5z5 - 11a5z7 + 4a5z9 + a6 - 6a6z2 + 13a6z4 - 11a6z6 + 3a6z8 - 2a7z + 5a7z3 - 4a7z5 + a7z7

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

{2, -3}

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 11169. 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                   5 1   j = 3                 8 3     j = 1               10 5       j = -1             10 9         j = -3           9 9           j = -5         7 10             j = -7       5 9               j = -9     2 7                 j = -11   1 5                   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, 169]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 169]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[18, 5, 19, 6], X[16, 7, 17, 8], > X[14, 10, 15, 9], X[2, 11, 3, 12], X[20, 14, 21, 13], X[22, 15, 1, 16], > X[6, 17, 7, 18], X[12, 20, 13, 19], X[8, 21, 9, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 169]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -9, 4, -11, 5, -2, 6, -10, 7, -5, 8, -4, 9, -3, 10, > -7, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 169]]
Out[5]=	DTCode[4, 10, 18, 16, 14, 2, 20, 22, 6, 12, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 169]][t]
Out[6]=	2 12 28 2 3 37 - -- + -- - -- - 28 t + 12 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 169]][z]
Out[7]=	2 6 1 + 2 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 36], Knot[11, Alternating, 169]}
In[9]:=	{KnotDet[Knot[11, Alternating, 169]], KnotSignature[Knot[11, Alternating, 169]]}
Out[9]=	{121, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 169]][q]
Out[10]=	-7 3 7 12 16 19 19 2 3 4 18 - q + -- - -- + -- - -- + -- - -- - 13 q + 8 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, 169]}
In[12]:=	A2Invariant[Knot[11, Alternating, 169]][q]
Out[12]=	-22 -18 3 2 -12 2 4 2 2 -2 2 4 -1 - q + q - --- + --- + q - --- + -- - -- + -- + q + 4 q - 4 q + 16 14 10 8 6 4 q q q q q q 6 8 10 12 > q + q - 2 q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 169]][a, z]
Out[13]=	2 4 2 4 6 2 z 2 2 4 2 6 2 4 z 2 4 a + a - a - 2 z + -- + a z + 3 a z - a z - 2 z + -- - a z + 2 2 a a 4 4 6 2 6 > 2 a z - z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 169]][a, z]
Out[14]=	2 2 4 6 5 7 2 2 z 2 2 4 2 6 2 -a + a + a - 2 a z - 2 a z + 2 z + ---- - 5 a z - 11 a z - 6 a z - 2 a 3 3 4 4 2 z 4 z 3 3 3 5 3 7 3 4 z 7 z > ---- + ---- + 8 a z + a z + 4 a z + 5 a z + 3 z + -- - ---- + 3 a 4 2 a a a 5 5 2 4 4 4 6 4 4 z 13 z 5 3 5 > 29 a z + 31 a z + 13 a z + ---- - ----- - 13 a z + 13 a z + 3 a a 6 7 5 5 7 5 6 8 z 2 6 4 6 6 6 11 z > 5 a z - 4 a z - 15 z + ---- - 40 a z - 28 a z - 11 a z + ----- - 2 a a 7 3 7 5 7 7 7 8 2 8 4 8 > 4 a z - 27 a z - 11 a z + a z + 11 z + 12 a z + 4 a z + 6 8 9 3 9 5 9 2 10 4 10 > 3 a z + 7 a z + 11 a z + 4 a z + 2 a z + 2 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 169]], Vassiliev[3][Knot[11, Alternating, 169]]}
Out[15]=	{2, -3}
In[16]:=	Kh[Knot[11, Alternating, 169]][q, t]
Out[16]=	9 1 2 1 5 2 7 5 9 - + 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 9 9 10 3 3 2 5 2 > ----- + ----- + ----- + ---- + --- + 5 q t + 8 q t + 3 q t + 5 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 K11a169

K11a168 K11a170