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

K11a92 K11a94

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 X20,13,21,14 X18,15,19,16 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, -8, 10, -7, 11, -5}

DT (Dowker-Thistlethwaite) Code:

4 10 12 16 22 2 20 18 8 14 6

Alexander Polynomial:

- 2t-3 + 10t-2 - 21t-1 + 27 - 21t + 10t2 - 2t3

Conway Polynomial:

1 + z2 - 2z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{10114, ...}

Determinant and Signature:

{93, 0}

Jones Polynomial:

- q-7 + 3q-6 - 6q-5 + 9q-4 - 12q-3 + 15q-2 - 14q-1 + 13 - 10q + 6q2 - 3q3 + q4

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

{...}

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- a-4z2 + a-4z4 + a-3z - 3a-3z3 + 3a-3z5 - a-2 + 3a-2z2 - 5a-2z4 + 5a-2z6 + a-1z + 2a-1z3 - 6a-1z5 + 6a-1z7 - 2 + 7z2 - 7z4 - 2z6 + 5z8 - az + 9az3 - 15az5 + 3az7 + 3az9 - 2a2 - a2z2 + 11a2z4 - 20a2z6 + 7a2z8 + a2z10 - 2a3z + 8a3z3 - 2a3z5 - 12a3z7 + 6a3z9 + a4 - 9a4z2 + 26a4z4 - 25a4z6 + 5a4z8 + a4z10 - 3a5z + 9a5z3 - 8a5z7 + 3a5z9 + a6 - 5a6z2 + 14a6z4 - 12a6z6 + 3a6z8 - 2a7z + 5a7z3 - 4a7z5 + a7z7

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

{1, -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 1193. 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                     2   j = 5                   4 1   j = 3                 6 2     j = 1               7 4       j = -1             8 7         j = -3           7 6           j = -5         5 8             j = -7       4 7               j = -9     2 5                 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, 93]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 93]]
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[20, 13, 21, 14], X[18, 15, 19, 16], > X[8, 17, 9, 18], X[14, 19, 15, 20], X[6, 22, 7, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 93]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -11, 4, -9, 5, -2, 6, -3, 7, -10, 8, -4, 9, -8, 10, > -7, 11, -5]
In[5]:=	DTCode[Knot[11, Alternating, 93]]
Out[5]=	DTCode[4, 10, 12, 16, 22, 2, 20, 18, 8, 14, 6]
In[6]:=	alex = Alexander[Knot[11, Alternating, 93]][t]
Out[6]=	2 10 21 2 3 27 - -- + -- - -- - 21 t + 10 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 93]][z]
Out[7]=	2 4 6 1 + z - 2 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 114], Knot[11, Alternating, 93]}
In[9]:=	{KnotDet[Knot[11, Alternating, 93]], KnotSignature[Knot[11, Alternating, 93]]}
Out[9]=	{93, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 93]][q]
Out[10]=	-7 3 6 9 12 15 14 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, 93]}
In[12]:=	A2Invariant[Knot[11, Alternating, 93]][q]
Out[12]=	-22 -18 2 -14 -10 4 2 2 4 6 8 10 -2 - q + q - --- + q - q + -- + -- + 2 q - 3 q + q + q - q + 16 8 4 q q q 12 > q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 93]][a, z]
Out[13]=	2 4 -2 2 4 6 2 2 z 4 2 6 2 4 z -2 + a + 2 a + a - a - 4 z + ---- + 4 a z - a z - 3 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, 93]][a, z]
Out[14]=	-2 2 4 6 z z 3 5 7 2 -2 - a - 2 a + a + a + -- + - - a z - 2 a z - 3 a z - 2 a z + 7 z - 3 a a 2 2 3 3 z 3 z 2 2 4 2 6 2 3 z 2 z 3 3 3 > -- + ---- - a z - 9 a z - 5 a z - ---- + ---- + 9 a z + 8 a z + 4 2 3 a a a a 4 4 5 3 7 3 4 z 5 z 2 4 4 4 6 4 > 9 a z + 5 a z - 7 z + -- - ---- + 11 a z + 26 a z + 14 a z + 4 2 a a 5 5 6 3 z 6 z 5 3 5 7 5 6 5 z 2 6 > ---- - ---- - 15 a z - 2 a z - 4 a z - 2 z + ---- - 20 a z - 3 a 2 a a 7 4 6 6 6 6 z 7 3 7 5 7 7 7 8 > 25 a z - 12 a z + ---- + 3 a z - 12 a z - 8 a z + a z + 5 z + a 2 8 4 8 6 8 9 3 9 5 9 2 10 4 10 > 7 a z + 5 a z + 3 a z + 3 a z + 6 a z + 3 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 93]], Vassiliev[3][Knot[11, Alternating, 93]]}
Out[15]=	{1, -3}
In[16]:=	Kh[Knot[11, Alternating, 93]][q, t]
Out[16]=	7 1 2 1 4 2 5 4 7 - + 7 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 5 8 7 6 8 3 3 2 5 2 > ----- + ----- + ----- + ---- + --- + 4 q t + 6 q t + 2 q t + 4 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 + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a93

K11a92 K11a94