10 1

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10_2

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 10 1's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 1 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X11,14,12,15 X3,13,4,12 X13,3,14,2 X5,20,6,1 X7,18,8,19 X9,16,10,17 X15,10,16,11 X17,8,18,9 X19,6,20,7
Gauss code	-1, 4, -3, 1, -5, 10, -6, 9, -7, 8, -2, 3, -4, 2, -8, 7, -9, 6, -10, 5
Dowker-Thistlethwaite code	4 12 20 18 16 14 2 10 8 6
Conway Notation	[82]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 13, width is 6, Braid index is 6		[{12, 9}, {8, 10}, {9, 7}, {6, 8}, {7, 5}, {4, 6}, {5, 3}, {2, 4}, {3, 1}, {11, 2}, {10, 12}, {1, 11}]

[edit Notes on presentations of 10 1]

Computer Talk[show]

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091. Read more at http://katlas.org/wiki/KnotTheory.

In[3]:=

K = Knot["10 1"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X11,14,12,15 X3,13,4,12 X13,3,14,2 X5,20,6,1 X7,18,8,19 X9,16,10,17 X15,10,16,11 X17,8,18,9 X19,6,20,7

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 12 20 18 16 14 2 10 8 6

(The path below may be different on your system)

In[7]:=

AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:=

ConwayNotation[K]

Out[8]=

[82]

In[9]:=

br = BR[K]

KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.

Out[9]=

${\textrm {BR}}(6,\{-1,-1,-2,1,-2,-3,2,-3,-4,3,5,-4,5\})$

In[10]:=

{First[br], Crossings[br], BraidIndex[K]}

KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.

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

Out[10]=

{ 6, 13, 6 }

In[11]:=

Show[BraidPlot[br]]

Out[11]=

-Graphics-

In[12]:=

Show[DrawMorseLink[K]]

KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."

KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

Out[12]=

-Graphics-

In[13]:=

ap = ArcPresentation[K]

Out[13]=

ArcPresentation[{12, 9}, {8, 10}, {9, 7}, {6, 8}, {7, 5}, {4, 6}, {5, 3}, {2, 4}, {3, 1}, {11, 2}, {10, 12}, {1, 11}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 1 Bridge index 2 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-9][-3] Hyperbolic Volume 3.5262 A-Polynomial See Data:10 1/A-polynomial

[edit Notes for 10 1's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $1$ Rasmussen s-Invariant 0

[edit Notes for 10 1's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-4t+9-4t^{-1}$
Conway polynomial	$1-4z^{2}$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 17, 0 }
Jones polynomial	$q^{2}-q+2-2q^{-1}+2q^{-2}-2q^{-3}+2q^{-4}-2q^{-5}+q^{-6}-q^{-7}+q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$a^{8}-z^{2}a^{6}-a^{6}-z^{2}a^{4}-z^{2}a^{2}-z^{2}+a^{-2}$
Kauffman polynomial (db, data sources)	$a^{7}z^{9}+a^{5}z^{9}+a^{8}z^{8}+2a^{6}z^{8}+a^{4}z^{8}-7a^{7}z^{7}-6a^{5}z^{7}+a^{3}z^{7}-7a^{8}z^{6}-12a^{6}z^{6}-4a^{4}z^{6}+a^{2}z^{6}+16a^{7}z^{5}+12a^{5}z^{5}-3a^{3}z^{5}+az^{5}+15a^{8}z^{4}+21a^{6}z^{4}+3a^{4}z^{4}-2a^{2}z^{4}+z^{4}-14a^{7}z^{3}-11a^{5}z^{3}+a^{3}z^{3}-az^{3}+z^{3}a^{-1}-10a^{8}z^{2}-11a^{6}z^{2}+z^{2}a^{-2}+4a^{7}z+4a^{5}z+a^{8}+a^{6}-a^{-2}$
The A2 invariant	$q^{26}+q^{24}-q^{18}-q^{16}+q^{-2}+q^{-6}+q^{-8}$
The G2 invariant	Data:10 1/QuantumInvariant/G2/1,0

Further Quantum Invariants[show]

Further quantum knot invariants for 10_1.

A1 Invariants.

Weight	Invariant
1	$q^{17}-q^{11}+q^{-1}+q^{-5}$
2	$q^{50}-q^{46}-q^{40}+q^{36}+q^{16}+q^{14}-q^{4}-q^{2}+q^{-2}+q^{-8}+q^{-14}$
3	$q^{99}-q^{95}-q^{93}+q^{89}-q^{85}+q^{81}+q^{79}-q^{75}+q^{49}+q^{47}-q^{43}-q^{37}-q^{35}-q^{29}+q^{25}+q^{23}+q^{15}+q^{13}+q^{11}-q^{9}+q^{5}+q^{3}-q-q^{-1}+q^{-3}+q^{-5}-q^{-7}-q^{-9}+q^{-19}+q^{-27}$

A2 Invariants.

Weight	Invariant
1,0	$q^{26}+q^{24}-q^{18}-q^{16}+q^{-2}+q^{-6}+q^{-8}$
2,0	$q^{68}+q^{66}+q^{64}-q^{62}-q^{60}-q^{58}-q^{56}-q^{54}-q^{52}+q^{50}+q^{48}+q^{46}+q^{24}+2q^{22}+q^{20}+q^{18}+q^{16}-q^{10}-2q^{8}-2q^{6}-q^{4}+q^{-4}+q^{-10}+q^{-12}+q^{-16}+q^{-18}+q^{-20}$

.

Computer Talk[show]

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 1"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-4t+9-4t^{-1}$

In[5]:=

Conway[K][z]

Out[5]=

$1-4z^{2}$

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]=

{ 17, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{2}-q+2-2q^{-1}+2q^{-2}-2q^{-3}+2q^{-4}-2q^{-5}+q^{-6}-q^{-7}+q^{-8}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$a^{8}-z^{2}a^{6}-a^{6}-z^{2}a^{4}-z^{2}a^{2}-z^{2}+a^{-2}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{7}z^{9}+a^{5}z^{9}+a^{8}z^{8}+2a^{6}z^{8}+a^{4}z^{8}-7a^{7}z^{7}-6a^{5}z^{7}+a^{3}z^{7}-7a^{8}z^{6}-12a^{6}z^{6}-4a^{4}z^{6}+a^{2}z^{6}+16a^{7}z^{5}+12a^{5}z^{5}-3a^{3}z^{5}+az^{5}+15a^{8}z^{4}+21a^{6}z^{4}+3a^{4}z^{4}-2a^{2}z^{4}+z^{4}-14a^{7}z^{3}-11a^{5}z^{3}+a^{3}z^{3}-az^{3}+z^{3}a^{-1}-10a^{8}z^{2}-11a^{6}z^{2}+z^{2}a^{-2}+4a^{7}z+4a^{5}z+a^{8}+a^{6}-a^{-2}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_3,}

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

Computer Talk[show]

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["10 1"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

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

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

Out[4]=

{ $-4t+9-4t^{-1}$, $q^{2}-q+2-2q^{-1}+2q^{-2}-2q^{-3}+2q^{-4}-2q^{-5}+q^{-6}-q^{-7}+q^{-8}$ }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{8_3,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{}

Vassiliev invariants

V2 and V3:

(-4, 6)

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 $-16$ $48$ $128$ ${\frac {232}{3}}$ ${\frac {200}{3}}$ $-768$ $-1088$ $-256$ $-272$ $-{\frac {2048}{3}}$ $1152$ $-{\frac {3712}{3}}$ $-{\frac {3200}{3}}$ ${\frac {30898}{15}}$ ${\frac {5768}{15}}$ $-{\frac {6248}{45}}$ ${\frac {4046}{9}}$ $-{\frac {4142}{15}}$

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

\   r    \    j   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 5                     1 1 3                       0 1                 2 1   1 -1               1 1     0 -3             1 1       0 -5           1 1         0 -7         1 1           0 -9       1 1             0 -11       1               -1 -13   1 1                 0 -15                       0 -17 1                     1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-8$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=1$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{6}-q^{5}+2q^{3}-2q^{2}+3-3q^{-1}-q^{-2}+3q^{-3}-2q^{-4}-q^{-5}+3q^{-6}-2q^{-7}+3q^{-9}-3q^{-10}+3q^{-12}-3q^{-13}+3q^{-15}-3q^{-16}+3q^{-18}-2q^{-19}-q^{-20}+2q^{-21}-q^{-22}-q^{-23}+q^{-24}$
3	$q^{12}-q^{11}+2q^{8}-2q^{7}+2q^{4}-3q^{3}+2q+2-5q^{-1}+4q^{-3}+2q^{-4}-6q^{-5}-q^{-6}+6q^{-7}+2q^{-8}-6q^{-9}-2q^{-10}+6q^{-11}+2q^{-12}-5q^{-13}-2q^{-14}+5q^{-15}+q^{-16}-4q^{-17}-2q^{-18}+4q^{-19}+q^{-20}-3q^{-21}-2q^{-22}+3q^{-23}+2q^{-24}-2q^{-25}-2q^{-26}+2q^{-27}+2q^{-28}-2q^{-29}-2q^{-30}+2q^{-31}+2q^{-32}-2q^{-33}-2q^{-34}+2q^{-35}+2q^{-36}-2q^{-37}-2q^{-38}+q^{-39}+3q^{-40}-q^{-41}-2q^{-42}+2q^{-44}-q^{-46}-q^{-47}+q^{-48}$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Rolfsen Knot Page master template (intermediate). See/edit the Rolfsen_Splice_Base (expert). Back to the top.

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