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

K11a91 K11a93

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 12 16 22 2 18 20 8 14 6

Alexander Polynomial:

- t-4 + 5t-3 - 12t-2 + 21t-1 - 25 + 21t - 12t2 + 5t3 - t4

Conway Polynomial:

1 + 2z2 - 2z4 - 3z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{103, -2}

Jones Polynomial:

- q-8 + 3q-7 - 7q-6 + 11q-5 - 14q-4 + 17q-3 - 16q-2 + 14q-1 - 10 + 6q - 3q2 + q3

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

{...}

A2 (sl(3)) Invariant:

- q-24 - q-20 - 2q-18 + 3q-16 - q-14 + 3q-12 + 2q-10 - q-8 + 3q-6 - 4q-4 + 2q-2 - 1 - q2 + 2q4 - q6 + q8

HOMFLY-PT Polynomial:

2 + 5z2 + 4z4 + z6 - 5a2 - 14a2z2 - 14a2z4 - 6a2z6 - a2z8 + 7a4 + 14a4z2 + 9a4z4 + 2a4z6 - 3a6 - 3a6z2 - a6z4

Kauffman Polynomial:

2a-2z2 - 3a-2z4 + a-2z6 - 2a-1z + 7a-1z3 - 9a-1z5 + 3a-1z7 + 2 - 5z2 + 7z4 - 10z6 + 4z8 - 5az + 12az3 - 10az5 - 2az7 + 3az9 + 5a2 - 23a2z2 + 39a2z4 - 30a2z6 + 8a2z8 + a2z10 - 6a3z + 12a3z3 + 3a3z5 - 13a3z7 + 7a3z9 + 7a4 - 27a4z2 + 47a4z4 - 34a4z6 + 10a4z8 + a4z10 - 6a5z + 12a5z3 - 5a5z5 - 3a5z7 + 4a5z9 + 3a6 - 10a6z2 + 13a6z4 - 12a6z6 + 6a6z8 - 2a7z + 3a7z3 - 8a7z5 + 5a7z7 + a8z2 - 5a8z4 + 3a8z6 + a9z - 2a9z3 + a9z5

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

{2, -5}

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 1192. 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 = 7                       1 j = 5                     2   j = 3                   4 1   j = 1                 6 2     j = -1               8 4       j = -3             9 7         j = -5           8 7           j = -7         6 9             j = -9       5 8               j = -11     2 6                 j = -13   1 5                   j = -15   2                     j = -17 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, 92]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 92]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[16, 7, 17, 8], > X[22, 10, 1, 9], X[2, 11, 3, 12], X[18, 13, 19, 14], X[20, 15, 21, 16], > X[8, 17, 9, 18], X[14, 19, 15, 20], X[6, 22, 7, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 92]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -11, 4, -9, 5, -2, 6, -3, 7, -10, 8, -4, 9, -7, 10, > -8, 11, -5]
In[5]:=	DTCode[Knot[11, Alternating, 92]]
Out[5]=	DTCode[4, 10, 12, 16, 22, 2, 18, 20, 8, 14, 6]
In[6]:=	alex = Alexander[Knot[11, Alternating, 92]][t]
Out[6]=	-4 5 12 21 2 3 4 -25 - t + -- - -- + -- + 21 t - 12 t + 5 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 92]][z]
Out[7]=	2 4 6 8 1 + 2 z - 2 z - 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 92]}
In[9]:=	{KnotDet[Knot[11, Alternating, 92]], KnotSignature[Knot[11, Alternating, 92]]}
Out[9]=	{103, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 92]][q]
Out[10]=	-8 3 7 11 14 17 16 14 2 3 -10 - q + -- - -- + -- - -- + -- - -- + -- + 6 q - 3 q + q 7 6 5 4 3 2 q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 92]}
In[12]:=	A2Invariant[Knot[11, Alternating, 92]][q]
Out[12]=	-24 -20 2 3 -14 3 2 -8 3 4 2 2 -1 - q - q - --- + --- - q + --- + --- - q + -- - -- + -- - q + 18 16 12 10 6 4 2 q q q q q q q 4 6 8 > 2 q - q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 92]][a, z]
Out[13]=	2 4 6 2 2 2 4 2 6 2 4 2 - 5 a + 7 a - 3 a + 5 z - 14 a z + 14 a z - 3 a z + 4 z - 2 4 4 4 6 4 6 2 6 4 6 2 8 > 14 a z + 9 a z - a z + z - 6 a z + 2 a z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 92]][a, z]
Out[14]=	2 4 6 2 z 3 5 7 9 2 2 + 5 a + 7 a + 3 a - --- - 5 a z - 6 a z - 6 a z - 2 a z + a z - 5 z + a 2 3 2 z 2 2 4 2 6 2 8 2 7 z 3 3 3 > ---- - 23 a z - 27 a z - 10 a z + a z + ---- + 12 a z + 12 a z + 2 a a 4 5 3 7 3 9 3 4 3 z 2 4 4 4 > 12 a z + 3 a z - 2 a z + 7 z - ---- + 39 a z + 47 a z + 2 a 5 6 4 8 4 9 z 5 3 5 5 5 7 5 9 5 > 13 a z - 5 a z - ---- - 10 a z + 3 a z - 5 a z - 8 a z + a z - a 6 7 6 z 2 6 4 6 6 6 8 6 3 z 7 > 10 z + -- - 30 a z - 34 a z - 12 a z + 3 a z + ---- - 2 a z - 2 a a 3 7 5 7 7 7 8 2 8 4 8 6 8 > 13 a z - 3 a z + 5 a z + 4 z + 8 a z + 10 a z + 6 a z + 9 3 9 5 9 2 10 4 10 > 3 a z + 7 a z + 4 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 92]], Vassiliev[3][Knot[11, Alternating, 92]]}
Out[15]=	{2, -5}
In[16]:=	Kh[Knot[11, Alternating, 92]][q, t]
Out[16]=	7 8 1 2 1 5 2 6 5 8 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 9 3 q q t q t q t q t q t q t q t q t 6 9 8 7 9 4 t 2 3 2 > ----- + ----- + ----- + ---- + ---- + --- + 6 q t + 2 q t + 4 q t + 7 3 7 2 5 2 5 3 q q t q t q t q t q t 3 3 5 3 7 4 > q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a92

K11a91 K11a93