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

K11a323 K11a325

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X12,3,13,4 X16,5,17,6 X22,8,1,7 X20,9,21,10 X18,11,19,12 X2,13,3,14 X8,15,9,16 X4,17,5,18 X10,19,11,20 X14,21,15,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 12 16 22 20 18 2 8 4 10 14

Alexander Polynomial:

- 5t-2 + 25t-1 - 39 + 25t - 5t2

Conway Polynomial:

1 + 5z2 - 5z4

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{99, -2}

Jones Polynomial:

- q-10 + 3q-9 - 7q-8 + 10q-7 - 13q-6 + 16q-5 - 15q-4 + 14q-3 - 10q-2 + 6q-1 - 3 + q

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

{...}

A2 (sl(3)) Invariant:

- q-32 - q-30 + 2q-28 - 2q-26 - q-24 + 3q-22 - 3q-20 + q-18 + q-16 + 3q-12 - q-10 + 3q-8 - 3q-4 + 3q-2 - 1 - q2 + q4

HOMFLY-PT Polynomial:

z2 + a2z2 - a2z4 + 2a4 + a4z2 - 2a4z4 - a6 - a6z2 - 2a6z4 + a8 + 3a8z2 - a10

Kauffman Polynomial:

- z2 + z4 - 3az3 + 3az5 + a2z2 - 5a2z4 + 5a2z6 + 7a3z3 - 11a3z5 + 7a3z7 + 2a4 - 11a4z2 + 22a4z4 - 19a4z6 + 8a4z8 + 2a5z - 4a5z3 + 10a5z5 - 13a5z7 + 6a5z9 + a6 - 21a6z2 + 45a6z4 - 35a6z6 + 6a6z8 + 2a6z10 + 7a7z - 23a7z3 + 38a7z5 - 34a7z7 + 10a7z9 + a8 - 12a8z2 + 28a8z4 - 22a8z6 + a8z8 + 2a8z10 + 3a9z - 4a9z3 + 10a9z5 - 13a9z7 + 4a9z9 + a10 - 4a10z2 + 11a10z4 - 11a10z6 + 3a10z8 - 2a11z + 5a11z3 - 4a11z5 + a11z7

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

{5, -12}

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 11324. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 3                       1 j = 1                     2   j = -1                   4 1   j = -3                 7 3     j = -5               7 3       j = -7             8 7         j = -9           8 7           j = -11         5 8             j = -13       5 8               j = -15     2 5                 j = -17   1 5                   j = -19   2                     j = -21 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, 324]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 324]]
Out[3]=	PD[X[6, 2, 7, 1], X[12, 3, 13, 4], X[16, 5, 17, 6], X[22, 8, 1, 7], > X[20, 9, 21, 10], X[18, 11, 19, 12], X[2, 13, 3, 14], X[8, 15, 9, 16], > X[4, 17, 5, 18], X[10, 19, 11, 20], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 324]]
Out[4]=	GaussCode[1, -7, 2, -9, 3, -1, 4, -8, 5, -10, 6, -2, 7, -11, 8, -3, 9, -6, 10, > -5, 11, -4]
In[5]:=	DTCode[Knot[11, Alternating, 324]]
Out[5]=	DTCode[6, 12, 16, 22, 20, 18, 2, 8, 4, 10, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 324]][t]
Out[6]=	5 25 2 -39 - -- + -- + 25 t - 5 t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 324]][z]
Out[7]=	2 4 1 + 5 z - 5 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 324]}
In[9]:=	{KnotDet[Knot[11, Alternating, 324]], KnotSignature[Knot[11, Alternating, 324]]}
Out[9]=	{99, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 324]][q]
Out[10]=	-10 3 7 10 13 16 15 14 10 6 -3 - q + -- - -- + -- - -- + -- - -- + -- - -- + - + q 9 8 7 6 5 4 3 2 q q 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, 324]}
In[12]:=	A2Invariant[Knot[11, Alternating, 324]][q]
Out[12]=	-32 -30 2 2 -24 3 3 -18 -16 3 -10 -1 - q - q + --- - --- - q + --- - --- + q + q + --- - q + 28 26 22 20 12 q q q q q 3 3 3 2 4 > -- - -- + -- - q + q 8 4 2 q q q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 324]][a, z]
Out[13]=	4 6 8 10 2 2 2 4 2 6 2 8 2 2 4 4 4 2 a - a + a - a + z + a z + a z - a z + 3 a z - a z - 2 a z - 6 4 > 2 a z
In[14]:=	Kauffman[Knot[11, Alternating, 324]][a, z]
Out[14]=	4 6 8 10 5 7 9 11 2 2 2 2 a + a + a + a + 2 a z + 7 a z + 3 a z - 2 a z - z + a z - 4 2 6 2 8 2 10 2 3 3 3 5 3 > 11 a z - 21 a z - 12 a z - 4 a z - 3 a z + 7 a z - 4 a z - 7 3 9 3 11 3 4 2 4 4 4 6 4 > 23 a z - 4 a z + 5 a z + z - 5 a z + 22 a z + 45 a z + 8 4 10 4 5 3 5 5 5 7 5 9 5 > 28 a z + 11 a z + 3 a z - 11 a z + 10 a z + 38 a z + 10 a z - 11 5 2 6 4 6 6 6 8 6 10 6 3 7 > 4 a z + 5 a z - 19 a z - 35 a z - 22 a z - 11 a z + 7 a z - 5 7 7 7 9 7 11 7 4 8 6 8 8 8 > 13 a z - 34 a z - 13 a z + a z + 8 a z + 6 a z + a z + 10 8 5 9 7 9 9 9 6 10 8 10 > 3 a z + 6 a z + 10 a z + 4 a z + 2 a z + 2 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 324]], Vassiliev[3][Knot[11, Alternating, 324]]}
Out[15]=	{5, -12}
In[16]:=	Kh[Knot[11, Alternating, 324]][q, t]
Out[16]=	3 4 1 2 1 5 2 5 5 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 21 9 19 8 17 8 17 7 15 7 15 6 13 6 q q t q t q t q t q t q t q t 8 5 8 8 7 8 7 7 3 > ------ + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 13 5 11 5 11 4 9 4 9 3 7 3 7 2 5 2 5 q t q t q t q t q t q t q t q t q t 7 t 3 2 > ---- + - + 2 q t + q t 3 q q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a324

K11a323 K11a325