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

K11a183 K11a185

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 14 16 18 22 2 20 8 10 6

Alexander Polynomial:

- t-4 + 5t-3 - 11t-2 + 17t-1 - 19 + 17t - 11t2 + 5t3 - t4

Conway Polynomial:

1 + 2z2 - z4 - 3z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{10112, ...}

Determinant and Signature:

{87, -2}

Jones Polynomial:

- q-8 + 3q-7 - 6q-6 + 9q-5 - 12q-4 + 14q-3 - 13q-2 + 12q-1 - 8 + 5q - 3q2 + q3

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

{...}

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

1 + 4z2 + 4z4 + z6 - 2a2 - 11a2z2 - 13a2z4 - 6a2z6 - a2z8 + 4a4 + 12a4z2 + 9a4z4 + 2a4z6 - 2a6 - 3a6z2 - a6z4

Kauffman Polynomial:

a-2z2 - 3a-2z4 + a-2z6 - a-1z + 7a-1z3 - 10a-1z5 + 3a-1z7 + 1 - 6z2 + 12z4 - 13z6 + 4z8 - 2az + 7az3 - 2az5 - 6az7 + 3az9 + 2a2 - 20a2z2 + 39a2z4 - 26a2z6 + 5a2z8 + a2z10 - a3z - 2a3z3 + 18a3z5 - 17a3z7 + 6a3z9 + 4a4 - 20a4z2 + 34a4z4 - 21a4z6 + 5a4z8 + a4z10 - a5z + 3a5z5 - 4a5z7 + 3a5z9 + 2a6 - 5a6z2 + 4a6z4 - 6a6z6 + 4a6z8 - 6a7z5 + 4a7z7 + 2a8z2 - 6a8z4 + 3a8z6 + a9z - 2a9z3 + a9z5

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

{2, -4}

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 11184. 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                   3 1   j = 1                 5 2     j = -1               7 3       j = -3             7 6         j = -5           7 6           j = -7         5 7             j = -9       4 7               j = -11     2 5                 j = -13   1 4                   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, 184]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 184]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 3, 13, 4], X[14, 6, 15, 5], X[16, 7, 17, 8], > X[18, 9, 19, 10], X[22, 12, 1, 11], X[2, 13, 3, 14], X[20, 15, 21, 16], > X[8, 17, 9, 18], X[10, 19, 11, 20], X[6, 22, 7, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 184]]
Out[4]=	GaussCode[1, -7, 2, -1, 3, -11, 4, -9, 5, -10, 6, -2, 7, -3, 8, -4, 9, -5, 10, > -8, 11, -6]
In[5]:=	DTCode[Knot[11, Alternating, 184]]
Out[5]=	DTCode[4, 12, 14, 16, 18, 22, 2, 20, 8, 10, 6]
In[6]:=	alex = Alexander[Knot[11, Alternating, 184]][t]
Out[6]=	-4 5 11 17 2 3 4 -19 - t + -- - -- + -- + 17 t - 11 t + 5 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 184]][z]
Out[7]=	2 4 6 8 1 + 2 z - z - 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 112], Knot[11, Alternating, 184]}
In[9]:=	{KnotDet[Knot[11, Alternating, 184]], KnotSignature[Knot[11, Alternating, 184]]}
Out[9]=	{87, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 184]][q]
Out[10]=	-8 3 6 9 12 14 13 12 2 3 -8 - q + -- - -- + -- - -- + -- - -- + -- + 5 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, 184]}
In[12]:=	A2Invariant[Knot[11, Alternating, 184]][q]
Out[12]=	-24 -18 2 2 -12 -10 4 2 2 2 4 6 8 -1 - q - q + --- - --- + q + q + -- - -- + -- - q + q - q + q 16 14 6 4 2 q q q q q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 184]][a, z]
Out[13]=	2 4 6 2 2 2 4 2 6 2 4 1 - 2 a + 4 a - 2 a + 4 z - 11 a z + 12 a z - 3 a z + 4 z - 2 4 4 4 6 4 6 2 6 4 6 2 8 > 13 a z + 9 a z - a z + z - 6 a z + 2 a z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 184]][a, z]
Out[14]=	2 2 4 6 z 3 5 9 2 z 1 + 2 a + 4 a + 2 a - - - 2 a z - a z - a z + a z - 6 z + -- - a 2 a 3 2 2 4 2 6 2 8 2 7 z 3 3 3 > 20 a z - 20 a z - 5 a z + 2 a z + ---- + 7 a z - 2 a z - a 4 5 9 3 4 3 z 2 4 4 4 6 4 8 4 10 z > 2 a z + 12 z - ---- + 39 a z + 34 a z + 4 a z - 6 a z - ----- - 2 a a 6 5 3 5 5 5 7 5 9 5 6 z 2 6 > 2 a z + 18 a z + 3 a z - 6 a z + a z - 13 z + -- - 26 a z - 2 a 7 4 6 6 6 8 6 3 z 7 3 7 5 7 > 21 a z - 6 a z + 3 a z + ---- - 6 a z - 17 a z - 4 a z + a 7 7 8 2 8 4 8 6 8 9 3 9 5 9 > 4 a z + 4 z + 5 a z + 5 a z + 4 a z + 3 a z + 6 a z + 3 a z + 2 10 4 10 > a z + a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 184]], Vassiliev[3][Knot[11, Alternating, 184]]}
Out[15]=	{2, -4}
In[16]:=	Kh[Knot[11, Alternating, 184]][q, t]
Out[16]=	6 7 1 2 1 4 2 5 4 7 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 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 5 7 7 6 7 3 t 2 3 2 > ----- + ----- + ----- + ---- + ---- + --- + 5 q t + 2 q t + 3 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 K11a184

K11a183 K11a185