10 3

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Visit 10 3's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 3's page at Knotilus! Visit 10 3's page at the original Knot Atlas! 10 3 Quick Notes

10 3 Further Notes and Views

Knot presentations

Planar diagram presentation	X1627 X11,16,12,17 X5,13,6,12 X3,15,4,14 X13,5,14,4 X15,3,16,2 X7,20,8,1 X9,18,10,19 X17,10,18,11 X19,8,20,9
Gauss code	-1, 6, -4, 5, -3, 1, -7, 10, -8, 9, -2, 3, -5, 4, -6, 2, -9, 8, -10, 7
Dowker-Thistlethwaite code	6 14 12 20 18 16 4 2 10 8
Conway Notation	[64]

Minimum Braid Representative: Length is 13, width is 6. Braid index is 6.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 1 Bridge index 2 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-7][-5] Hyperbolic Volume 5.7321 A-Polynomial See Data:10 3/A-polynomial

[edit Notes for 10 3's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $0$ Topological 4 genus $0$ Concordance genus $0$ Rasmussen s-Invariant 0

[edit Notes for 10 3's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-6t+13-6t^{-1}$
Conway polynomial	$1-6z^{2}$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 25, 0 }
Jones polynomial	$q^{4}-q^{3}+2q^{2}-3q+4-4q^{-1}+3q^{-2}-3q^{-3}+2q^{-4}-q^{-5}+q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$a^{6}-z^{2}a^{4}-2z^{2}a^{2}-a^{2}-2z^{2}-z^{2}a^{-2}+a^{-4}$
Kauffman polynomial (db, data sources)	$a^{3}z^{9}+az^{9}+a^{4}z^{8}+2a^{2}z^{8}+z^{8}+a^{5}z^{7}-6a^{3}z^{7}-6az^{7}+z^{7}a^{-1}+a^{6}z^{6}-4a^{4}z^{6}-10a^{2}z^{6}+z^{6}a^{-2}-4z^{6}-4a^{5}z^{5}+15a^{3}z^{5}+15az^{5}-3z^{5}a^{-1}+z^{5}a^{-3}-5a^{6}z^{4}+4a^{4}z^{4}+18a^{2}z^{4}-2z^{4}a^{-2}+z^{4}a^{-4}+6z^{4}+3a^{5}z^{3}-18a^{3}z^{3}-15az^{3}+4z^{3}a^{-1}-2z^{3}a^{-3}+6a^{6}z^{2}-2a^{4}z^{2}-12a^{2}z^{2}+z^{2}a^{-2}-3z^{2}a^{-4}+6a^{3}z+6az-a^{6}+a^{2}+a^{-4}$
The A2 invariant	$q^{20}+q^{18}+q^{14}-q^{10}-q^{6}-q^{4}+q^{-2}-q^{-4}+q^{-8}+q^{-12}+q^{-14}$
The G2 invariant	Data:10 3/QuantumInvariant/G2/1,0

Further Quantum Invariants

Further quantum knot invariants for 10_3.

A1 Invariants.

Weight	Invariant
1	$q^{13}+q^{9}-q^{7}-q^{3}+q^{-1}-q^{-3}+q^{-5}+q^{-9}$
2	$q^{38}+q^{32}-q^{30}-2q^{28}+q^{26}-2q^{22}+q^{20}+q^{18}-q^{16}+2q^{12}+q^{6}+2q^{-2}-q^{-4}-q^{-6}+2q^{-8}-q^{-10}-q^{-12}-q^{-16}+q^{-20}+q^{-26}$
3	$q^{75}-q^{65}-q^{63}+q^{59}-q^{57}-2q^{55}+3q^{51}+2q^{49}-2q^{47}-2q^{45}+q^{43}+3q^{41}-q^{37}-q^{35}+q^{33}+2q^{31}+q^{29}-3q^{27}-q^{25}+2q^{23}+2q^{21}-2q^{19}-2q^{17}+q^{15}-q^{11}-q^{9}+q^{7}+q^{3}-q-q^{-1}+2q^{-3}+2q^{-5}-2q^{-9}+3q^{-13}+q^{-15}-2q^{-19}+2q^{-23}+2q^{-25}-q^{-27}-2q^{-29}+q^{-33}-2q^{-37}-q^{-39}+q^{-43}+q^{-51}$

A2 Invariants.

Weight	Invariant
1,0	$q^{20}+q^{18}+q^{14}-q^{10}-q^{6}-q^{4}+q^{-2}-q^{-4}+q^{-8}+q^{-12}+q^{-14}$
2,0	$q^{52}+q^{50}+q^{48}+q^{44}+q^{42}-q^{40}-3q^{38}-2q^{36}-2q^{30}-q^{28}+q^{26}+q^{24}-q^{20}+2q^{18}+2q^{16}+2q^{10}+q^{8}+q^{6}+q^{4}+1+q^{-2}+q^{-4}-2q^{-6}-2q^{-8}+2q^{-10}-3q^{-14}-2q^{-16}-q^{-22}+q^{-26}+q^{-28}+q^{-32}+q^{-34}+q^{-36}$

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["10 3"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$-6t+13-6t^{-1}$

In[5]:=

Conway[K][z]

Out[5]=

$1-6z^{2}$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 25, 0 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$q^{4}-q^{3}+2q^{2}-3q+4-4q^{-1}+3q^{-2}-3q^{-3}+2q^{-4}-q^{-5}+q^{-6}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$a^{6}-z^{2}a^{4}-2z^{2}a^{2}-a^{2}-2z^{2}-z^{2}a^{-2}+a^{-4}$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$a^{3}z^{9}+az^{9}+a^{4}z^{8}+2a^{2}z^{8}+z^{8}+a^{5}z^{7}-6a^{3}z^{7}-6az^{7}+z^{7}a^{-1}+a^{6}z^{6}-4a^{4}z^{6}-10a^{2}z^{6}+z^{6}a^{-2}-4z^{6}-4a^{5}z^{5}+15a^{3}z^{5}+15az^{5}-3z^{5}a^{-1}+z^{5}a^{-3}-5a^{6}z^{4}+4a^{4}z^{4}+18a^{2}z^{4}-2z^{4}a^{-2}+z^{4}a^{-4}+6z^{4}+3a^{5}z^{3}-18a^{3}z^{3}-15az^{3}+4z^{3}a^{-1}-2z^{3}a^{-3}+6a^{6}z^{2}-2a^{4}z^{2}-12a^{2}z^{2}+z^{2}a^{-2}-3z^{2}a^{-4}+6a^{3}z+6az-a^{6}+a^{2}+a^{-4}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {...}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {...}

Vassiliev invariants

V2 and V3:

(-6, 3)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $-24$ $24$ $288$ $372$ $148$ $-576$ $-912$ $-192$ $-200$ $-2304$ $288$ $-8928$ $-3552$ $-{\frac {36071}{5}}$ ${\frac {21692}{15}}$ $-{\frac {99844}{15}}$ ${\frac {2551}{3}}$ $-{\frac {7751}{5}}$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). 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 10 3. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 9                     1 1 7                       0 5                 2 1   1 3               1       -1 1             3 2       1 -1           2 2         0 -3         1 2           -1 -5       2 2             0 -7       1               -1 -9   1 2                 1 -11                       0 -13 1                     1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=3$ ${\mathbb {Z} }$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the $n+1$-dimensional representation of $sl(2)$)

$n$	$J_{n}$
2	$q^{12}-q^{11}+2q^{9}-2q^{8}-q^{7}+3q^{6}-3q^{5}-q^{4}+6q^{3}-6q^{2}-q+9-8q^{-1}-q^{-2}+9q^{-3}-7q^{-4}-2q^{-5}+9q^{-6}-5q^{-7}-4q^{-8}+8q^{-9}-3q^{-10}-4q^{-11}+5q^{-12}-q^{-13}-3q^{-14}+2q^{-15}-q^{-17}+q^{-18}$
3	$q^{24}-q^{23}+2q^{20}-2q^{19}-q^{18}-q^{17}+4q^{16}-q^{15}-2q^{14}-3q^{13}+5q^{12}+2q^{11}-2q^{10}-5q^{9}+3q^{8}+4q^{7}-q^{6}-3q^{5}+2q^{3}+q^{2}-q-q^{-1}+q^{-2}+q^{-3}-q^{-4}-q^{-6}+q^{-7}+q^{-9}-4q^{-10}+q^{-11}+4q^{-12}+q^{-13}-7q^{-14}-q^{-15}+8q^{-16}+2q^{-17}-8q^{-18}-3q^{-19}+8q^{-20}+3q^{-21}-5q^{-22}-5q^{-23}+5q^{-24}+3q^{-25}-q^{-26}-4q^{-27}+2q^{-28}+q^{-29}-2q^{-31}+q^{-32}-q^{-35}+q^{-36}$
4	Not Available
5	Not Available
6	Not Available
7	Not Available

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
In[2]:=	PD[Knot[10, 3]]
Out[2]=	PD[X[1, 6, 2, 7], X[11, 16, 12, 17], X[5, 13, 6, 12], X[3, 15, 4, 14], X[13, 5, 14, 4], X[15, 3, 16, 2], X[7, 20, 8, 1], X[9, 18, 10, 19], X[17, 10, 18, 11], X[19, 8, 20, 9]]
In[3]:=	GaussCode[Knot[10, 3]]
Out[3]=	GaussCode[-1, 6, -4, 5, -3, 1, -7, 10, -8, 9, -2, 3, -5, 4, -6, 2, -9, 8, -10, 7]
In[4]:=	DTCode[Knot[10, 3]]
Out[4]=	DTCode[6, 14, 12, 20, 18, 16, 4, 2, 10, 8]
In[5]:=	br = BR[Knot[10, 3]]
Out[5]=	BR[6, {-1, -1, -2, 1, -2, -3, 2, 4, -3, 4, 5, -4, 5}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{6, 13}
In[7]:=	BraidIndex[Knot[10, 3]]
Out[7]=	6
In[8]:=	Show[DrawMorseLink[Knot[10, 3]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[10, 3]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 1, 2, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 3]][t]
Out[10]=	6 13 - - - 6 t t
In[11]:=	Conway[Knot[10, 3]][z]
Out[11]=	2 1 - 6 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 3]}
In[13]:=	{KnotDet[Knot[10, 3]], KnotSignature[Knot[10, 3]]}
Out[13]=	{25, 0}
In[14]:=	Jones[Knot[10, 3]][q]
Out[14]=	-6 -5 2 3 3 4 2 3 4 4 + q - q + -- - -- + -- - - - 3 q + 2 q - q + q 4 3 2 q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 3]}
In[16]:=	A2Invariant[Knot[10, 3]][q]
Out[16]=	-20 -18 -14 -10 -6 -4 2 4 8 12 14 q + q + q - q - q - q + q - q + q + q + q
In[17]:=	HOMFLYPT[Knot[10, 3]][a, z]
Out[17]=	2 -4 2 6 2 z 2 2 4 2 a - a + a - 2 z - -- - 2 a z - a z 2 a
In[18]:=	Kauffman[Knot[10, 3]][a, z]
Out[18]=	2 2 -4 2 6 3 3 z z 2 2 4 2 a + a - a + 6 a z + 6 a z - ---- + -- - 12 a z - 2 a z + 4 2 a a 3 3 4 6 2 2 z 4 z 3 3 3 5 3 4 z 6 a z - ---- + ---- - 15 a z - 18 a z + 3 a z + 6 z + -- - 3 a 4 a a 4 5 5 2 z 2 4 4 4 6 4 z 3 z 5 ---- + 18 a z + 4 a z - 5 a z + -- - ---- + 15 a z + 2 3 a a a 6 7 3 5 5 5 6 z 2 6 4 6 6 6 z 15 a z - 4 a z - 4 z + -- - 10 a z - 4 a z + a z + -- - 2 a a 7 3 7 5 7 8 2 8 4 8 9 3 9 6 a z - 6 a z + a z + z + 2 a z + a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 3]], Vassiliev[3][Knot[10, 3]]}
Out[19]=	{-6, 3}
In[20]:=	Kh[Knot[10, 3]][q, t]
Out[20]=	2 1 1 2 1 2 2 1 - + 3 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 9 5 9 4 7 3 5 3 5 2 3 2 q t q t q t q t q t q t q t 2 2 3 5 2 5 3 9 4 ---- + --- + 2 q t + q t + 2 q t + q t + q t 3 q t q t
In[21]:=	ColouredJones[Knot[10, 3], 2][q]
Out[21]=	-18 -17 2 3 -13 5 4 3 8 4 5 9 + q - q + --- - --- - q + --- - --- - --- + -- - -- - -- + 15 14 12 11 10 9 8 7 q q q q q q q q 9 2 7 9 -2 8 2 3 4 5 6 -- - -- - -- + -- - q - - - q - 6 q + 6 q - q - 3 q + 3 q - 6 5 4 3 q q q q q 7 8 9 11 12 q - 2 q + 2 q - q + q

See/edit the Rolfsen_Splice_Template. Back to the top.

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