# 10 123

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 10 123's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 123 at Knotilus! 10_123 can be depicted with five-fold rotational symmetry (like 5 1).

 Quasi-floral decorative knot. Decorative pentagonal representation. Symmetrical "flower". Cylindrical depiction

### Knot presentations

 Planar diagram presentation X8291 X10,3,11,4 X12,6,13,5 X4,18,5,17 X18,11,19,12 X2,15,3,16 X16,10,17,9 X20,14,1,13 X14,7,15,8 X6,19,7,20 Gauss code 1, -6, 2, -4, 3, -10, 9, -1, 7, -2, 5, -3, 8, -9, 6, -7, 4, -5, 10, -8 Dowker-Thistlethwaite code 8 10 12 14 16 18 20 2 4 6 Conway Notation [10*]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 10, width is 3,

Braid index is 3

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

### Three dimensional invariants

 Symmetry type Fully amphicheiral Unknotting number 2 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index 2 Maximal Thurston-Bennequin number [-6][-6] Hyperbolic Volume 17.0857 A-Polynomial See Data:10 123/A-polynomial

### Four dimensional invariants

 Smooth 4 genus ${\displaystyle 0}$ Topological 4 genus ${\displaystyle 0}$ Concordance genus ${\displaystyle 0}$ Rasmussen s-Invariant 0

### Polynomial invariants

 Alexander polynomial ${\displaystyle t^{4}-6t^{3}+15t^{2}-24t+29-24t^{-1}+15t^{-2}-6t^{-3}+t^{-4}}$ Conway polynomial ${\displaystyle z^{8}+2z^{6}-z^{4}-2z^{2}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \left\{t^{4}-3t^{3}+3t^{2}-3t+1\right\}}$ Determinant and Signature { 121, 0 } Jones polynomial ${\displaystyle -q^{5}+5q^{4}-10q^{3}+15q^{2}-19q+21-19q^{-1}+15q^{-2}-10q^{-3}+5q^{-4}-q^{-5}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle z^{8}-a^{2}z^{6}-z^{6}a^{-2}+4z^{6}-2a^{2}z^{4}-2z^{4}a^{-2}+3z^{4}+a^{2}z^{2}+z^{2}a^{-2}-4z^{2}+2a^{2}+2a^{-2}-3}$ Kauffman polynomial (db, data sources) ${\displaystyle 4az^{9}+4z^{9}a^{-1}+10a^{2}z^{8}+10z^{8}a^{-2}+20z^{8}+10a^{3}z^{7}+14az^{7}+14z^{7}a^{-1}+10z^{7}a^{-3}+5a^{4}z^{6}-11a^{2}z^{6}-11z^{6}a^{-2}+5z^{6}a^{-4}-32z^{6}+a^{5}z^{5}-15a^{3}z^{5}-38az^{5}-38z^{5}a^{-1}-15z^{5}a^{-3}+z^{5}a^{-5}-5a^{4}z^{4}-3a^{2}z^{4}-3z^{4}a^{-2}-5z^{4}a^{-4}+4z^{4}+5a^{3}z^{3}+21az^{3}+21z^{3}a^{-1}+5z^{3}a^{-3}+6a^{2}z^{2}+6z^{2}a^{-2}+12z^{2}-2az-2za^{-1}-2a^{2}-2a^{-2}-3}$ The A2 invariant ${\displaystyle -q^{14}+3q^{12}-2q^{10}+3q^{8}-3q^{4}+4q^{2}-5+4q^{-2}-3q^{-4}+3q^{-8}-2q^{-10}+3q^{-12}-q^{-14}}$ The G2 invariant ${\displaystyle q^{80}-4q^{78}+10q^{76}-20q^{74}+26q^{72}-25q^{70}+10q^{68}+30q^{66}-80q^{64}+140q^{62}-180q^{60}+158q^{58}-71q^{56}-100q^{54}+308q^{52}-473q^{50}+528q^{48}-391q^{46}+69q^{44}+343q^{42}-693q^{40}+822q^{38}-656q^{36}+239q^{34}+267q^{32}-647q^{30}+750q^{28}-495q^{26}+29q^{24}+435q^{22}-675q^{20}+551q^{18}-133q^{16}-414q^{14}+836q^{12}-944q^{10}+710q^{8}-173q^{6}-472q^{4}+970q^{2}-1165+970q^{-2}-472q^{-4}-173q^{-6}+710q^{-8}-944q^{-10}+836q^{-12}-414q^{-14}-133q^{-16}+551q^{-18}-675q^{-20}+435q^{-22}+29q^{-24}-495q^{-26}+750q^{-28}-647q^{-30}+267q^{-32}+239q^{-34}-656q^{-36}+822q^{-38}-693q^{-40}+343q^{-42}+69q^{-44}-391q^{-46}+528q^{-48}-473q^{-50}+308q^{-52}-100q^{-54}-71q^{-56}+158q^{-58}-180q^{-60}+140q^{-62}-80q^{-64}+30q^{-66}+10q^{-68}-25q^{-70}+26q^{-72}-20q^{-74}+10q^{-76}-4q^{-78}+q^{-80}}$

### "Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a28,}

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

### Vassiliev invariants

 V2 and V3: (-2, 0)
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 ${\displaystyle -8}$ ${\displaystyle 0}$ ${\displaystyle 32}$ ${\displaystyle {\frac {164}{3}}}$ ${\displaystyle {\frac {76}{3}}}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle -{\frac {256}{3}}}$ ${\displaystyle 0}$ ${\displaystyle -{\frac {1312}{3}}}$ ${\displaystyle -{\frac {608}{3}}}$ ${\displaystyle -{\frac {6271}{15}}}$ ${\displaystyle {\frac {1484}{15}}}$ ${\displaystyle -{\frac {16684}{45}}}$ ${\displaystyle {\frac {31}{9}}}$ ${\displaystyle -{\frac {1471}{15}}}$

V2,1 through V6,9 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 ${\displaystyle t^{r}q^{j}}$ are shown, along with their alternating sums ${\displaystyle \chi }$ (fixed ${\displaystyle j}$, alternation over ${\displaystyle r}$). The squares with yellow highlighting are those on the "critical diagonals", where ${\displaystyle j-2r=s+1}$ or ${\displaystyle j-2r=s-1}$, where ${\displaystyle s=}$0 is the signature of 10 123. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-5-4-3-2-1012345χ
11          1-1
9         4 4
7        61 -5
5       94  5
3      106   -4
1     119    2
-1    911     2
-3   610      -4
-5  49       5
-7 16        -5
-9 4         4
-111          -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-1}$ ${\displaystyle i=1}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}}$ ${\displaystyle {\mathbb {Z} }^{9}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}}$ ${\displaystyle {\mathbb {Z} }^{11}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}}$ ${\displaystyle {\mathbb {Z} }^{10}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}}$ ${\displaystyle {\mathbb {Z} }^{9}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$