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

K11a37 K11a39

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X14,5,15,6 X2837 X18,9,19,10 X16,11,17,12 X20,14,21,13 X6,15,7,16 X10,17,11,18 X22,19,1,20 X12,22,13,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 14 2 18 16 20 6 10 22 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:

{K11a8, K11a187, K11a249, ...}

Determinant and Signature:

{117, 0}

Jones Polynomial:

q-6 - 3q-5 + 7q-4 - 12q-3 + 16q-2 - 19q-1 + 19 - 16q + 13q2 - 7q3 + 3q4 - q5

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

{...}

A2 (sl(3)) Invariant:

q-18 - q-16 + 2q-14 + 2q-12 - 4q-10 + 2q-8 - 3q-6 - 2q-4 + 2q-2 - 2 + 5q2 - q4 + 2q6 + 3q8 - 3q10 + q12 - q16

HOMFLY-PT Polynomial:

- a-4 - a-4z2 + 2a-2 + 3a-2z2 + 2a-2z4 + 2 + z2 - z4 - z6 - 4a2 - 6a2z2 - 3a2z4 - a2z6 + 2a4 + 2a4z2 + a4z4

Kauffman Polynomial:

a-5z - 2a-5z3 + a-5z5 - a-4 + 3a-4z2 - 5a-4z4 + 3a-4z6 + a-3z + a-3z3 - 6a-3z5 + 5a-3z7 - 2a-2 + 3a-2z4 - 7a-2z6 + 6a-2z8 + 7a-1z3 - 11a-1z5 + 2a-1z7 + 4a-1z9 + 2 - 17z2 + 36z4 - 34z6 + 13z8 + z10 + 5az3 - 2az5 - 10az7 + 8az9 + 4a2 - 23a2z2 + 43a2z4 - 38a2z6 + 12a2z8 + a2z10 - 2a3z + 7a3z3 - 6a3z5 - 4a3z7 + 4a3z9 + 2a4 - 7a4z2 + 12a4z4 - 13a4z6 + 5a4z8 - 2a5z + 6a5z3 - 8a5z5 + 3a5z7 + 2a6z2 - 3a6z4 + a6z6

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 1138. 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 = 11                       1 j = 9                     2   j = 7                   5 1   j = 5                 8 2     j = 3               8 5       j = 1             11 8         j = -1           9 9           j = -3         7 10             j = -5       5 9               j = -7     2 7                 j = -9   1 5                   j = -11   2                     j = -13 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, 38]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 38]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[14, 5, 15, 6], X[2, 8, 3, 7], > X[18, 9, 19, 10], X[16, 11, 17, 12], X[20, 14, 21, 13], X[6, 15, 7, 16], > X[10, 17, 11, 18], X[22, 19, 1, 20], X[12, 22, 13, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 38]]
Out[4]=	GaussCode[1, -4, 2, -1, 3, -8, 4, -2, 5, -9, 6, -11, 7, -3, 8, -6, 9, -5, 10, > -7, 11, -10]
In[5]:=	DTCode[Knot[11, Alternating, 38]]
Out[5]=	DTCode[4, 8, 14, 2, 18, 16, 20, 6, 10, 22, 12]
In[6]:=	alex = Alexander[Knot[11, Alternating, 38]][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, 38]][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, 38]], KnotSignature[Knot[11, Alternating, 38]]}
Out[9]=	{117, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 38]][q]
Out[10]=	-6 3 7 12 16 19 2 3 4 5 19 + q - -- + -- - -- + -- - -- - 16 q + 13 q - 7 q + 3 q - q 5 4 3 2 q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 38]}
In[12]:=	A2Invariant[Knot[11, Alternating, 38]][q]
Out[12]=	-18 -16 2 2 4 2 3 2 2 2 4 6 -2 + q - q + --- + --- - --- + -- - -- - -- + -- + 5 q - q + 2 q + 14 12 10 8 6 4 2 q q q q q q q 8 10 12 16 > 3 q - 3 q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 38]][a, z]
Out[13]=	2 2 4 -4 2 2 4 2 z 3 z 2 2 4 2 4 2 z 2 - a + -- - 4 a + 2 a + z - -- + ---- - 6 a z + 2 a z - z + ---- - 2 4 2 2 a a a a 2 4 4 4 6 2 6 > 3 a z + a z - z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 38]][a, z]
Out[14]=	2 -4 2 2 4 z z 3 5 2 3 z 2 - a - -- + 4 a + 2 a + -- + -- - 2 a z - 2 a z - 17 z + ---- - 2 5 3 4 a a a a 3 3 3 2 2 4 2 6 2 2 z z 7 z 3 3 3 > 23 a z - 7 a z + 2 a z - ---- + -- + ---- + 5 a z + 7 a z + 5 3 a a a 4 4 5 5 5 3 4 5 z 3 z 2 4 4 4 6 4 z 6 z > 6 a z + 36 z - ---- + ---- + 43 a z + 12 a z - 3 a z + -- - ---- - 4 2 5 3 a a a a 5 6 6 11 z 5 3 5 5 5 6 3 z 7 z 2 6 > ----- - 2 a z - 6 a z - 8 a z - 34 z + ---- - ---- - 38 a z - a 4 2 a a 7 7 4 6 6 6 5 z 2 z 7 3 7 5 7 8 > 13 a z + a z + ---- + ---- - 10 a z - 4 a z + 3 a z + 13 z + 3 a a 8 9 6 z 2 8 4 8 4 z 9 3 9 10 2 10 > ---- + 12 a z + 5 a z + ---- + 8 a z + 4 a z + z + a z 2 a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 38]], Vassiliev[3][Knot[11, Alternating, 38]]}
Out[15]=	{-1, 3}
In[16]:=	Kh[Knot[11, Alternating, 38]][q, t]
Out[16]=	9 1 2 1 5 2 7 5 9 - + 11 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 7 10 9 3 3 2 5 2 5 3 > ----- + ---- + --- + 8 q t + 8 q t + 5 q t + 8 q t + 2 q t + 3 2 3 q t q t q t 7 3 7 4 9 4 11 5 > 5 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a38

K11a37 K11a39